∎

11institutetext:  P. Cumani 22institutetext: CSNSM, CNRS and University of
Paris Sud, F-91405, Orsay, France
22email<EMAIL_ADDRESS>33institutetext: M. Hernanz
44institutetext: Institute of Space Sciences (ICE, CSIC) and IEEC, Campus UAB,
Camí de Can Magrans s/n, 08193 Cerdanyola del Vallès (Barcelona), Spain
55institutetext: J. Kiener, V. Tatischeff66institutetext: CSNSM, CNRS and
University of Paris Sud, F-91405, Orsay, France
77institutetext: A. Zoglauer 88institutetext: University of California at
Berkeley, Space Sciences Laboratory, 7 Gauss Way, Berkeley, CA 94720, USA

# Background for a gamma-ray satellite on a low-Earth orbit

P. Cumani  M. Hernanz  J. Kiener  V. Tatischeff  A. Zoglauer

(Received: date / Accepted: date)

###### Abstract

The different background components in a low-Earth orbit have been modeled in
the 10 keV to 100 GeV energy range. The model is based on data from previous
instruments and it considers both primary and secondary particles, charged
particles, neutrons and photons. The necessary corrections to consider the
geomagnetic cutoff are applied to calculate the flux at different inclinations
and altitudes for a mean solar activity. Activation simulations from such a
background have been carried out using the model of a possible future gamma-
ray mission (e-ASTROGAM). The event rates and spectra from these simulations
were then compared to those from the isotopes created by the particles present
in the South Atlantic Anomaly (SAA). The primary protons are found to be the
main contributor of the activation, while the contributions of the neutrons,
and that of the secondary protons can be considered negligible. The long-term
activation from the passage through the SAA becomes the main source of
background at high inclination (i$\gtrsim 10^{\circ}$). The used models have
been collected in a Python class openly available on github.

###### Keywords:

Background Gamma-ray satellite LEO

††journal: Experimental Astronomy

##  1 Introduction

Conventionally, Low-Earth Orbits (LEOs) have a maximum altitude of $\sim 2000$
km, with a $\sim 300$ km lower limit given by the rapid orbital decay caused
by the atmospheric drag. The inner Van Allen radiation belt (van Allen
(1958)), with a high level of trapped radiation, extends beyond $\sim 1000$
km. Satellites positioned at lower altitudes benefit, at low geomagnetic
latitudes, of the shielding of the Earth’s magnetic field, blocking a large
part of the charged particles coming from the interplanetary medium.
As a drawback, all LEOs are subject to the drag caused by the remaining
atmosphere and to the background from albedo and secondary particles created
in the atmosphere.

Focusing on the characteristics of a gamma-ray mission in a low inclination
orbit, a model for each component of the background in the 10 keV to 100 GeV
energy range will be described in section 2. The South Atlantic Anomaly (SAA),
the area in which the inner Van Allen belt comes closer to Earth, will be
illustrated in section 3, simulating the particle spectrum along a typical
orbit at an altitude of 550 km. The computed spectra will be used in section 4
to simulate the activation of a possible future gamma-ray satellite,
e-ASTROGAM, in order to estimate the contribution of the different components
to the overall background and the importance of the radioisotopes created
during the passage of the SAA for different inclinations. Conclusions are
given in section 5.

##  2 Background models

Figure 1: Variation of the average geomagnetic cutoff $R_{cutoff}$ with
altitude for constant latitude (0∘, red dashed line), and latitude for
constant altitude (550 km, blue straight line).

The LEO background spectrum depends mainly on the orbit altitude and
inclination as well as the current solar activity.
In the following, where not differently specified, an altitude $h=550$ km,
inclination of the orbit $i=0^{\circ}$, and a solar modulation potential value
of 650 MV, corresponding to an activity halfway between minimum and maximum,
will be used. The magnetic latitude is approximated to be equal, on average,
to the inclination of the orbit.
Some of the parameters that are used to calculate more than one component are
computed in the following way:

  * •

Horizon angle (in degree): defined as the angle between the zenith and the top
of the atmosphere ($H_{A}=40$ km from sea level) at the satellite altitude
$h$:

$\theta_{H}=90^{\circ}+\arccos\frac{R_{\Earth}+H_{A}}{R_{\Earth}+h}$ (1)

where $R_{\Earth}$ is the Earth’s radius. The horizon angle for a 550 km
equatorial orbit is equal to $112^{\circ}$.

  * •

Average geomagnetic cutoff (in GV): is calculated as in Smart and Shea (2005):

$R_{cutoff}=\frac{M_{d}\cdot\cos^{4}\lambda}{r^{2}}\cdot\frac{1}{(1+\sqrt{1-\sin\epsilon\sin\xi\cos^{3}\lambda})^{2}}$
(2)

where $M_{d}$ is the magnitude of Earth’s dipole moment, $\lambda$ is the
latitude from the magnetic equator, $\epsilon$ is the angle from the zenith
direction, $\xi$ is the azimuthal angle measured clockwise from the direction
to the north magnetic pole and $r$ is the distance from the dipole center. $r$
can be expressed as $R_{\Earth}+h$. Since we are computing the omnidirectional
flux, $\xi$ can be set to zero. $M_{d}$ can be calculated by multiplying the
$g^{0}_{1}$ term of the International Geomagnetic Reference Field (IGRF) by
the cube of the average radius of the Earth. The value from the
IGRF-12111https://www.ngdc.noaa.gov/IAGA/vmod/igrf.html for the year 2015
($g^{0}_{1}=29442.0$ nT) has been used222It must be noted that this value
changes slowly, 0.5% in the last 10 years, but constantly due to the evolution
of Earth’s magnetic field. Equation 2 can then be simplified to:

$\begin{split}R_{cutoff}&=\frac{g^{0}_{1}\cdot
R_{\Earth}}{4}\cdot\left(1+\frac{h}{R_{\Earth}}\right)^{-2}\cos^{4}\lambda\\\
&=11.9~{}\mathrm{GV}\cdot\frac{g^{0}_{1}}{29442.0~{}\mathrm{nT}}\cdot\left(\frac{R_{\Earth}+550~{}\mathrm{km}}{R_{\Earth}+h}\right)^{2}\cos^{4}\lambda\end{split}$
(3)

As shown in Fig. 1, the average geomagnetic cutoff varies more steeply with
geomagnetic latitude than altitude.

In the following, the results of all the flux equations will be expressed as
counts keV-1 cm-2 s-1 sr-1. All the models of the different components
described in the following have been collected in a Python class in an open
github repository333https://github.com/pcumani/LEOBackground.

###  2.1 Extragalactic Photons

Figure 2: Total extragalactic photons flux as modeled by INTEGRAL-IBIS/ISGRI,
at low energies, and FERMI-LAT, at high energy.

The isotropic extragalactic X and gamma-ray emission is both a significant
component of the background for observation far from the Galactic plane, and
an important science topic, especially in the MeV energy range (see, e.g., De
Angelis et al (2018)).
The spectra shown in Fig. 2 are obtained using:

  * •

$E<890$ keV: From an extrapolation of the equation in Türler et al (2010),
describing data of the IBIS/ISGRI gamma-ray imager instrument of INTEGRAL
(Winkler et al (2003)):

$F=\frac{0.109}{(E/28~{}\mathrm{keV})^{1.4}+(E/28~{}\mathrm{keV})^{2.88}}$ (4)

  * •

$E\geqslant 890$ keV: Using the Fermi-LAT (Atwood et al (2009)) results of the
fit for the so-called foreground model A from Ackermann et al (2015) and
extrapolating it to lower energies from $E\geqslant 100$ MeV:

$\begin{split}F&=I_{100}\left(\frac{E}{100~{}\mathrm{MeV}}\right)^{-\gamma}\exp{\frac{-E}{E_{cut}}}\\\
&=0.95\cdot
10^{-10}\left(\frac{E}{100~{}\mathrm{MeV}}\right)^{-2.32}\exp{\left(\frac{-E}{279\cdot
10^{3}~{}\mathrm{MeV}}\right)}\end{split}$ (5)

The extrapolated Fermi-LAT data are in good agreement with EGRET data
(Sreekumar et al (1998), see equation 18 from Mizuno et al (2004)).

###  2.2 Galactic plane

Figure 3: Extragalactic photon spectrum compared with the background spectrum
from the Galactic center and the Galactic disk (top), and the ratio between
these last two (bottom). Figure 4: The two regions defined as Galactic center
(in red) and Galactic disk (in green) in the text, superimposed on the
publicly available Fermi-LAT background at 58 MeV. The inset shows a zoom on
the Galactic center region

It’s useful, especially at high energy, to study separately the gamma-ray
background for observations of the Galactic plane region. Such background
extends up to the highest considered energies and, starting from $\sim 200$
MeV, is more than an order of magnitude larger than the isotropic background
previously described.
It’s worth noting that the signal-to-noise ratio is directly related to the
performance of the instruments. Thanks to a higher angular resolution, new
instruments could be able to discriminate several point sources in previously
confused areas, either Galactic or not.
Currently, the most sensitive gamma-ray experiment in the considered energy
range is the Fermi-LAT. The Pass 8 interstellar emission
model444https://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html
was used to obtain the results shown in Fig 3. For each one of the 30 energies
provided, starting from $\sim 58$ MeV, two regions has been defined as shown
in Fig. 4:

  * •

Galactic center: the most inner part of the Galaxy, defined by the coordinate
$-1^{\circ}<b<1^{\circ}$, $-2.5^{\circ}<l<2.5^{\circ}$.

  * •

Galactic disk: a region defined as $-2^{\circ}<b<2^{\circ}$,
$-90^{\circ}<l<90^{\circ}$, excluding the $-2.5^{\circ}<l<2.5^{\circ}$ area
around the Galactic center.

The average flux in these two regions was computed at each energy. The ratio
between the two regions varies between a factor of 2.4 to 3.4, increasing with
energy, as shown at the bottom of Fig. 3.

###  2.3 Albedo Photons

Figure 5: Total albedo photons spectrum divided by the solid angle of the
atmosphere ($\sim 4$ sr at 550 km). The different models composing it are also
showed (see text).

Albedo photons are created by the Earth atmosphere, as a result of the
interaction of cosmic rays or the reflection of the cosmic X-ray background.
To model the zenith dependence of the spectrum, following the approach used by
Mizuno et al (2004), an angular distribution peaking at the top of the
atmosphere ($\sim 112^{\circ}$ at 550 km), to reproduce the limb-brightening
observed in experimental data (see, e.g., Ackermann et al (2014)), was used.
The flux gradually decreases at higher angles reaching 50% of its peak value
at $\sim 130^{\circ}$ (for an altitude of 550 km) and a minimum, one third of
the maximum, at the nadir.
The results shown in Fig. 5 are obtained using:

  * •

$E<1.85$ MeV: The sum of two components, both used in the analysis of INTEGRAL
observations of the Earth, resulting in a good agreement between data and
prediction:

    * •

To describe the hard X-ray surface brightness of the Earth’s atmosphere we
used the output of equation 1 from Sazonov et al (2007):

$F=\frac{C}{(E/44~{}\mathrm{keV})^{-5}+(E/44~{}\mathrm{keV})^{1.4}}$ (6)

where $C$ is:

$C=\frac{3\mu(1+\mu)}{5\pi}\cdot\frac{1.47\cdot
0.0178[(\phi/2.8)^{0.4}+(\phi/2.8)^{1.5}]^{-1}}{\sqrt{1+\\{R_{cutoff}/[1.3\phi^{0.25}(1+2.5\phi^{0.4})]\\}^{2}}}$
(7)

where $\mu=cos(180-\theta_{H})$ is the cosine of the zenith angle and $\phi$
is the solar modulation potential in GV.

    * •

the reflected X-ray background $F=F_{0}\cdot\Omega\cdot A(E)$, where $A(E)$ is
calculated with equation 9 in Churazov et al (2006):

$\begin{split}A(E)&=\frac{1.22}{\left(\frac{E}{28.5}\right)^{-2.54}+\left(\frac{E}{51.3}\right)^{1.57}-0.37}\\\
&\cdot\frac{2.93+\left(\frac{E}{3.08}\right)^{4}}{1+\left(\frac{E}{3.08}\right)^{4}}\\\
&\cdot\frac{0.123+\left(\frac{E}{91.83}\right)^{3.44}}{1+\left(\frac{E}{91.83}\right)^{3.44}}\end{split}$
(8)

being E the energy in keV, and $F_{0}$ with equation 4 from section 2.1.

  * •

1.85 MeV $<E<$ 200 MeV: from SAS 2 data (Thompson et al (1981)) as described
in Mizuno et al (2004):

    * •

$E<20$ MeV:

$F=\frac{1010}{10^{7}}\cdot\left(\frac{E}{1\mathrm{MeV}}\right)^{-1.34}\cdot\left(\frac{R_{cutoff}}{4.5~{}\mathrm{GV}}\right)^{-1.13}$
(9)

    * •

20 MeV $<E<$ 200 MeV:

$F=\frac{7290}{10^{7}}\cdot\left(\frac{E}{1~{}\mathrm{MeV}}\right)^{-2}\cdot\left(\frac{R_{cutoff}}{4.5~{}\mathrm{GV}}\right)^{-1.13}$
(10)

  * •

$E>200$ MeV: The high energy component is taken from Fermi-LAT data (Abdo et
al (2009b)):

$F=1.823\cdot 10^{-8}\cdot\left(\frac{E}{200\mathrm{MeV}}\right)^{-2.8}$ (11)

The INTEGRAL and Fermi-LAT data are normalized to the results of equations 9
and 10 at, respectively, 1.85 and 200 MeV, to take into account the effect of
the geomagnetic cutoff.

###  2.4 Primary Protons

Figure 6: Spectrum of the primary charged particles: protons, alphas,
electrons, and positrons. Figure 7: Ratio between the model presented here and
the results by SPENVIS, obtained using different models for the galactic
cosmic rays (CREME96 and ISO 15390) in different solar activity condition
(minimum and maximum) and two different methods for computing the rigidity
cut-off (Størmer with eccentric dipole model, the default option, and Smart
and Shea (1967) with the IGRF model, the latter of which was used only with
the CREME96 model).

Cosmic rays consist mainly of protons, a lot of which are trapped by the
geomagnetic field and never reach LEO. AMS data (Aguilar et al (2015b)) for
the unmodulated proton spectrum have been used, together with the reduction
factor from the geomagnetic cutoff described in Mizuno et al (2004), to model
the component:

$F=F_{p,AMS}\cdot\frac{1}{1+(R/R_{cutoff})^{-12}}$ (12)

where $R$ is the proton rigidity and $R_{cutoff}$ is defined in equation 3.
Results are shown in Fig. 6.
While results obtained by tools like ESA’s SPace ENVironment Information
System (SPENVIS555https://www.spenvis.oma.be/) are more realistic, by
considering, e.g., the variation of $R_{cutoff}$ along the orbit, the approach
described by eq. 12 was chosen for consistency with the approach used to
calculate the primary electrons/positrons spectra (that can not be calculated
using SPENVIS), and to allow the use of a single tool to calculate all the
background components for different LEOs. A comparison with the results from
SPENVIS for an equatorial orbit at 550 km is shown in Fig. 7. The results are
shown only in the energy range provided by SPENVIS (1 MeV$/n<E<10$0 GeV$/n$)
and are calculated using the two available cosmic-ray models: CREME96 for a
solar minimum and ISO 15390 for both minimum and maximum solar activity. When
using the default description of the geomagnetic field, the differential
spectrum is null for 1 MeV$<E\lesssim 7$GeV in case of protons, and 1
MeV$/n<E\lesssim 3$GeV$/n$ for the alpha particles. Differences up to a factor
of 2 are visible around the value of the geomagnetic cutoff. By changing the
method used by SPENVIS to compute the rigidity cut-
off666https://www.spenvis.oma.be/help/models/magshielding.html the ratio with
the presented model reaches a maximum factor of 1.5 at the energy of the
cutoff. For energies higher than the cutoff SPENVIS predicts a higher flux. It
should be noted that such differences have little effect on the detector
activation due to the creation of unstable isotopes, which is dominated by the
higher part of the spectrum.

###  2.5 Primary Alpha Particles

A similar approach as the one used for the protons has been used also for the
primary alpha particles. The unmodulated spectrum was derived from AMS data
(Aguilar et al (2015a)), to which the reduction factor from the geomagnetic
cutoff was lately applied:

$F=F_{\alpha,AMS}\cdot\frac{1}{1+(R/R_{cutoff})^{-12}}$ (13)

where $R$ is the particle rigidity and $R_{cutoff}$ is defined in equation 3.
Again, as per the primary protons, while the SPENVIS results are more
realistic, the described approach was followed for consistency. As per the
protons, and with similar results, a comparison with the results from SPENVIS
for an equatorial orbit at 550 km is shown in Fig. 7.

###  2.6 Primary Electrons

Consistently with what has been done for the protons and alpha particles, the
primary spectrum from AMS data (Aguilar et al (2014)) is used, with the
subsequent application of the geomagnetic cutoff modulations:

$F=F_{e,AMS}\cdot\frac{1}{1+(R/R_{cutoff})^{-6}}$ (14)

Although the geomagnetic cutoff is expected to be the same regardless of the
particle type, a reduction factor of the form $1/1+(R/R_{cutoff})^{-n}$, with
$n=6$, gives a better description of the AMS-01 (Alcaraz et al (2000a)) data
for electrons and positrons than with $n=12$ (see Mizuno et al (2004)). The
resulting flux is shown in Fig. 6.

###  2.7 Primary Positrons

Exactly like for the primary electrons, a combination of AMS data (Aguilar et
al (2014)) and the subsequent application of the same geomagnetic reduction
factor with $n=6$ (eq. 14), were used. The results are shown in Fig. 6.

###  2.8 Secondary Protons

Figure 8: Spectra of on-orbit secondary protons describing both the upward and
downward component for two different geomagnetic cutoffs: $R_{cutoff}=11.9$ GV
(the one, e.g. for a 550 km equatorial orbit) and $R_{cutoff}=11.2$ GV (e.g.,
for a 550 km orbit at an inclination of 10∘).

When cosmic rays interact with the atmosphere’s particles, they produce
secondary particles. The secondary protons, shown in Fig. 8, are computed
following Mizuno et al (2004), based on AMS-01 data (Alcaraz et al (2000b)):

  * •

1 MeV $<E<$ 100 MeV: Equation 9 of Mizuno et al (2004)

$F=\frac{0.136}{10^{7}}\cdot\left(\frac{E}{100~{}\mathrm{MeV}}\right)^{-1}$
(15)

is extrapolated down to 1 MeV. Below 1 MeV there is no probability for a
proton of creating an unstable isotope which decay could add to the total
background.

  * •

E $>$ 100 MeV:

$F=\frac{0.123}{10^{7}}\cdot\left(\frac{E}{1000~{}\mathrm{MeV}}\right)^{-0.155}\exp{-\left(\frac{E}{510~{}\mathrm{MeV}}\right)^{-0.845}}$
(16)

Mizuno et al (2004) model the secondary proton flux at the altitude of AMS-01
(380 km) as a function of the geomagnetic latitude $\lambda$. We assume that
the geomagnetic cutoff $R_{cutoff}$ is a more relevant parameter to describe
the particle flux for different altitudes and inclinations. Equations 15 and
16 describe both the upward and downward fluxes of the secondary protons in
high geomagnetic cutoff regions ($R_{cutoff}\gtrsim 11.5$ GV). At lower cutoff
values, down to $\sim 10.4$ GV, the spectra of the downward and upward
components take the shape of a broken power-law at energies greater than 100
MeV (Mizuno et al (2004)):

  * •

100 MeV $<E<$ 600 MeV:

$F=\frac{0.1}{10^{7}}\left(\frac{E}{100~{}\mathrm{MeV}}\right)^{-0.87}$ (17)

  * •

$E\geqslant$ 600 MeV:

$F=\frac{0.1}{10^{7}}\left(\frac{600}{100}\right)^{-0.87}\left(\frac{E}{600~{}\mathrm{MeV}}\right)^{-2.53}$
(18)

The two components assume different shapes at lower cutoff values (see Mizuno
et al (2004)).

###  2.9 Secondary Electrons and Positrons

Figure 9: Secondary electrons and positrons spectra.

The spectra of the secondary electrons and positrons, shown in Fig. 9, are
based on data from AMS-01 (Alcaraz et al (2000a)), modeled as in Mizuno et al
(2004), extrapolating their equations to 1 MeV. For the electrons the
equations are:

  * •

1 MeV $<E<$ 100 MeV:

$F=\frac{0.3}{10^{7}}\cdot\left(\frac{E}{100~{}\mathrm{MeV}}\right)^{-1}$ (19)

  * •

100 MeV $<E<$ 3 GeV:

$F=\frac{0.3}{10^{7}}\cdot\left(\frac{E}{100~{}\mathrm{MeV}}\right)^{-2.2}$
(20)

  * •

$E>$ 3 GeV

$F=\frac{0.3}{10^{7}}\cdot\left(\frac{E}{100~{}\mathrm{MeV}}\right)^{-2.2}\left(\frac{E}{3~{}\mathrm{GeV}}\right)^{-4}$
(21)

For the positrons the same equations are multiplied by a factor 3.33 from the
e${}^{+}/$e- ratio in equatorial orbit. As per the secondary protons, the
spectra of the secondary electrons and positrons depends on the geomagnetic
cutoff. In particular, the equations above are valid down to a cutoff value of
$\sim 10.4$ GV, corresponding to an inclination of $\sim 15^{\circ}$ at an
altitude of 550 km.

###  2.10 Atmospheric Neutrons

Figure 10: On-orbit spectrum (out of the atmosphere) of the cosmic-ray induced
atmospheric neutrons.

When interacting with the atmosphere, galactic cosmic-rays can create hadronic
showers. Neutrons created in such showers can reach instrument on LEOs.
As reported in Kole et al (2015), since the downward component tends to zero
at high altitudes, the on-orbit atmospheric neutrons spectrum is calculated
following the description of only the upward component.
The flux can be described by a power-law $F=x_{i}\cdot E^{-y_{i}}\cdot
10^{-3}$ with four different normalization and exponents in four different
energy intervals (Kole et al (2015)). The equations describing $x_{i}$ and
${y_{i}}$ depend on the geomagnetic latitude $\lambda$ and the solar activity
parameter $S=(\phi-250~{}\mathrm{MV})/(859~{}\mathrm{MV})$, where $\phi$ is
the solar modulation potential ($S=0.47$ for $\phi=650$ MV). These equations
are:

  * •

E $<$ 0.9 MeV:

$\begin{split}&a=3\cdot 10^{-4}+(7-5\cdot S)\cdot
10^{-3}[1-\tanh(180-4\cdot\lambda)]\\\ &b=1.4\cdot 10^{-2}+(1.4-0.9\cdot
S)\cdot 10^{-1}[1-\tanh(180-3.5\cdot\lambda)]\\\
&c=180-42\cdot[1-\tanh(180-5.5\cdot\lambda)]\\\ &d=-8\cdot 10^{-3}+(6-S)\cdot
10^{-3}[1-\tanh(180-4.4\cdot\lambda)]\\\ &x_{1}=(a\cdot p+b)e^{-p/c}+d\\\
&y_{1}=-0.29e^{-p/7.5}+0.735\end{split}$ (22)

  * •

0.9 MeV $<$ E $<$ 15 MeV:

$\begin{split}&x_{2}=x_{1}\cdot 0.9^{-y_{1}+y_{2}}\\\
&y_{2}=-0.247e^{-p/36.5}+1.4\end{split}$ (23)

  * •

15 MeV $<$ E $<$ 70 MeV:

$\begin{split}&x_{3}=x_{2}\cdot 15^{-y_{2}+y_{3}}\\\
&y_{3}=-0.4e^{-p/40}+0.9\end{split}$ (24)

  * •

E $>$ 70 MeV:

$\begin{split}&x_{4}=x_{3}\cdot 70^{-y_{3}+y_{4}}\\\
&y_{4}=-0.46e^{-p/100}+2.53\end{split}$ (25)

The results are shown in Fig. 10. As shown in Kole et al (2015), this model is
found to be in good agreement with data from high altitude aircraft, balloon-
borne experiments and previously published simulations.
The model developed by Kole et al (2015) is valid down to 8 keV. The spectrum
at the top of the atmosphere from Lingenfelter (1963), scaled to the Kole et
al (2015) results, was used to describe the spectrum between 0.01 eV and 8
keV. The Lingenfelter (1963) results outside the atmosphere for energies lower
than 1 MeV are compatible with the calculations in Armstrong et al (1973),
which in turns are compatible with the results in Kole et al (2015).
To compute the result shown in Fig. 10, the result of the previously described
equations has been divided by
$\Omega=4\pi-2\pi(1-cos\theta_{H})=2\pi(1+cos\theta_{H})$ to get a flux per
unit steradian.

###  2.11 Total Spectrum

Figure 11: Full on-orbit background spectrum for charged particles and
neutrons (top) and photons (bottom). All spectra components shown are divided
by the solid angle of their region of origin. The pointing and field-of-view
of an hypothetical instrument are therefore not considered

Figure 12: Ratio between fluxes obtained for different orbits: at the
variation of the altitude, from 550 km to 600 km (top) or inclination, from
$0^{\circ}$ to $10^{\circ}$ (bottom).

A collection of all the results for a 550 km equatorial orbit is shown in Fig.
11. Up to $\sim 100$ keV, the dominant component of the photon background
spectrum is represented by photons of extragalactic origins, subsequently
overtaken by gamma rays generated in the atmosphere. Depending on the line of
sight, the main component of the background can become the Galactic center,
starting from $\sim 400$ keV, or the Galactic disk, starting from 1 GeV, with
the albedo photons being still a non negligible component up to the highest
energy.
Given the big uncertainty due to a lack of data and the variation due to the
solar modulation, for energies lower than 1 MeV only the neutron component is
calculated. Secondary charged particles, positrons first and protons later,
give the highest contribution up to $\sim 6$ GeV, where they are surpassed by
the primary protons. While most of the particles interaction with the
satellite should be rejected by an anticoincidence system (99.97% in the case
of Fermi Atwood et al (2009)), some of them could interact with part of the
detector and create long living isotopes which decay could resemble a Compton
event. A study on the background activation, using the spectrum of Fig. 11, is
presented in the next section.
At the change of either altitude or orbit inclination, both shown in Fig. 12,
the spectra of some of the components remain unchanged. The photons
background, either Galactic or extra-Galactic, does not change in any way with
the orbit, regardless of the model used. The spectra of secondary charged
particles (protons, electrons, and positrons), as modeled here, depends solely
on the geomagnetic cutoff which, as demonstrated before, varies more with the
inclination than with the altitude (see Fig.2). For a small change of altitude
no changes are seen in any of the secondary components. The secondary protons
spectra however change its shape at higher inclination leading to a decrease
in the spectrum up to a factor $>2$ from $\sim$300 MeV up to $\sim 4$ GeV,
with a subsequent rapid increase. For such a change in inclination (0∘ to
10∘), the secondary electrons and positrons spectra remain unchanged. The
albedo photons spectrum as shown in Fig. 5 and 11 changes both with the
inclination and the altitude of the orbit. The variation is due to the
dependence on the geomagnetic cutoff and, at low energies, a more complicated
dependence on the Earth viewing angle. A difference of $10\%$ or less is found
when changing to 600 km maintaining the same inclination, or to $10^{\circ}$
keeping the altitude constant. Up to 200 keV, in the energy range dominated by
the reflected extragalactic X-ray background, dependent only on the Earth
viewing angle, a different trend with up to 7% reduction at increasing
altitude can be observed.
On LEOs, the atmospheric neutrons spectrum is more dependent on the
inclination than on the altitude. It changes by $\sim 17\%$ at the variation
of the inclination, and less than 10% when changing altitude.
All primary charged particles spectra are dependent on both altitude and
inclination of the orbit through their dependence on the geomagnetic cutoff
$R_{cutoff}$. The differences in the spectra thus appear at energies lower
than the cutoff, quickly disappearing at the highest energy. For protons and
alpha particles, the spectrum of which is modulated by a factor
$R_{cutoff}^{-12}$, this difference can reach a factor of 2 increasing the
inclination or 20% while changing the altitude. Electrons and positrons
spectra are more loosely dependent on the geomagnetic cutoff, only by a factor
$R_{cutoff}^{-6}$, leading to a maximum difference of a factor $\sim 1.4$.
However these differences appear in an energy range where the particle fluxes
are strongly reduced by the geomagnetic modulation, making them not important
in the background calculation.

| Orbit Parameters | Energy
---|---|---
Extragalactic Photons | Independent | 4 keV - 820 GeV
Galactic Photons | Independent | 58 MeV - $\sim 513$ GeV
Albedo Photons | All LEOs | 1 keV - 400 GeV
Primary Protons | All LEOs | 10 MeV - 10 TeV
Primary Alphas | All LEOs | 10 MeV - 10 TeV
Primary Electrons/Positrons | All LEOs | 570 MeV - 429 GeV
Secondary Protons | $1.06\leq R_{cutoff}\leq 12.47$ | 1 MeV - 10 GeV
Secondary Electrons/Positrons | $1.06\leq R_{cutoff}\leq 12.47$ | 1 MeV - 20 GeV
Atmospheric Neutrons | Altitude: $\sim 100$ km - $\sim 1000$ km | 0.01 eV - $\sim 30$ GeV
Inclination: $<65^{\circ}$
Table 1: Validity limits of the different components of the background.

The validity limits of the model for the different components of the
background are presented in Tab. 1.

##  3 South Atlantic Anomaly

Figure 13: Top: Average particle flux experienced in one month by a satellite
passing through the South Atlantic Anomaly, as calculated by the AE9/AP9-IRENE
model. Bottom: Ratio of the spectrum as calculated for different inclinations
with respect to the spectrum on an equatorial orbit ($i=0^{\circ}$) Figure 14:
Ratio of the trapped particles flux at different altitudes calculated for
different orbit inclinations. Similarly to Fig. 13, the different extensions
of the spectra at higher energies create a sharp increase in the ratio. It
should be noted that at these energies the fluxes are already negligible.

The South Atlantic Anomaly (SAA) is the region where the inner Van Allen
radiation belt comes closest to the Earth surface. Satellites passing through
this area are subject to intense radiation, caused mainly by protons and
electrons. Scientific observations are usually stopped while passing through
the SAA, although some diagnostic data can still be recorded (see e.g. Abdo et
al (2009a)). It is possible for protons, interacting with the satellite, to
create short and long lived radioisotopes, which would decay after resuming
observation, adding to the total background. To simulate the creation and
decay of such isotopes, the first step is to estimate the SAA spectrum along a
given orbit.
The AE9/AP9-IRENE (Ginet et al (2013)) models (version 1.5, O’Brien et al
(2018)) have been used to simulate the SAA passage at an altitude of 550 km
and along differently inclined
($0^{\circ},1^{\circ},2^{\circ},3^{\circ},4^{\circ},5^{\circ},10^{\circ}$)
orbits. The model allowed to compute the mean, omnidirectional, flux along the
orbit with a 10 s time step. The results for a total simulated time of 1 month
($\sim 455$ orbits at 550 km) are shown in Fig. 13, together with a ratio of
the values at different inclinations. The average flux increases with the
orbit inclination, in a similar way for both electrons and protons. The
difference with respect to the equatorial orbit reaches a factor of more than
6 for an inclination of $10^{\circ}$, and about an order of magnitude for
$i=15^{\circ}$. At greater inclinations, a greater difference between the low
($<10$ MeV) and high energy part of the proton spectrum can also be noted.
With an increase of altitude the flux increases, more at lower orbit
inclinations than at higher ones, as shown in Fig. 14. To an increase in
altitude of 50 km, it corresponds an increment of the flux between a factor of
$\sim 2$ and $\sim 3$ for inclinations lower than $5^{\circ}$.
A low inclination and low altitude orbit ensures the lowest possible flux
during the passage through the SAA. A perfectly equatorial orbit would grant
the best conditions, but little changes are observed for inclination smaller
than 3∘. To this kind of orbit corresponds also the smallest average time
passed by the satellite inside the SAA. For a 550 km equatorial orbit,
considering particles with $E>0.1$ MeV, the time corresponds to around 18% of
the orbit period (95 minutes at 550 km). An increase larger than 20% is seen
for a 600 km orbit, while differences up to 20% are observed with varying
inclinations.

##  4 Activation Simulations

Isotope | T1/2 | Activationi / Activationmax | Volumes
---|---|---|---
General | SAA
Background | 1 yr | 1 orb.
11C | 20 m | 0.2 (0.6 sp) | 1 | 0.8 | Anticoincidence
Supports
Spacecraft
Electronics
|  |  |  |  | Calorimeter Crystals
128I | 25 m | 1 | 0.9 | 0.4
126I | 13 d | 0.3 | 0.7 | $<0.1$
132Cs | 6.5 d | 0.2 | 0.8 | $<0.1$
134Cs | 2.1 yr | 0.5 | 0.3 | $<0.1$
125I | 59 d | 0.2 | 0.5 | $<0.1$
131Cs | 9.7 d | 0.1 | 0.5 | $<0.1$
123I | 13 h | 0.1 | 0.5 | $<0.1$
122I | 3.6 m | 0.1 | 0.1 | 0.1
126Cs | 1.6 m | $<0.1$ | $<0.1$ | 0.1
25Al | 7.2 s | $<0.1$ (0.1 sp) | $<0.1$ | 0.5 | Spacecraft
Electronics
Tracker Wafers
Calorimeter Diodes
28Al | 2.2 m | 0.3 (0.4 n) | $<0.1$ | 0.5 | Calorimeter Diodes
Electronics
Tracker Wafers
15O | 2 m | 0.1 (0.3 sp) | $<0.1$ | 1 | Anticoincidence
Electronics
Supports
Table 2: List of the most relevant isotopes created by the particles passing
through the different sub-systems of the e-ASTROGAM satellite model. Results
are given as relative to the most active source of background, i.e. 128I for
the general background, 11C for SAA-1 yr and 15O for the SAA short-term
activation. The results are generally consistent for all the components, some
exception to this are observed for the atmospheric neutrons (n) and secondary
protons (sp).

The obtained spectra can be used as input to simulate the detector activation
due to hadron interactions (Zoglauer et al (2008); Dyer et al (1989);
Battersby et al (1993)). The simulations were performed using the Medium
Energy Gamma-ray Astronomy library (MEGAlib) toolkit (Zoglauer et al (2006)).
They are divided in three steps:

  1. 1.

Simulation and storage of the isotopes generated by the passage of the initial
particle through the considered spacecraft or material.

  2. 2.

Calculation of the activation radiation after a certain irradiation time. The
irradiation can either be constant during the whole time, or be followed by a
certain cool down time.

  3. 3.

Simulation of the isotopes decay after the irradiation.

The spacecraft used in the simulation is a mass model of the gamma-ray mission
proposal e-ASTROGAM (De Angelis et al (2017)).
Dedicated to the gamma-ray observation in the 300 keV to 3 GeV energy range,
the e-ASTROGAM instrument will consist of three main detectors: a tracker, a
calorimeter and an anticoincidence system. The tracker is foreseen to be
composed of 56 silicon layers supported on their side by a mechanical
structure. The calorimeter is composed by thallium activated cesium iodide
crystals, read out by silicon drift detectors. The anticoincidence, covering
both the top and the side of the instrument and used to veto the triggers from
charged particles, is made by plastic scintillator coupled to silicon
photomultipliers. Thanks to this geometry, e-ASTROGAM will be able to
reconstruct gamma-ray events in both the Compton and pair regime.

###  4.1 Simulations

The simulated particles for the activation simulations were: primary and
secondary protons, alpha particles, atmospheric neutrons, and protons trapped
in the SAA.
For the primary and secondary protons, as well as for the atmospheric neutrons
and alpha particles, from which the irradiation can be considered steady along
a given orbit, a constant irradiation for 1 year was simulated. The spectrum
used was the one described in sec. 2, supposing an equatorial orbit with 550
km of altitude.
In the case of the SAA protons, since with MEGAlib it is not yet possible to
simulate the variation of the spectrum at the passage of the satellite through
the area, the average spectrum of a one month observation, shown in Fig. 13
was used. The short and long term activation were studied separately in the
case of the SAA. For the short term, the one affecting observation at every
orbit, just after the passage through the anomaly, simulations were carried
out for 10 minutes of constant irradiation with some cool down time (0, 1, 2,
5, 10 minutes). The cool down corresponds to a time after the end of the
irradiation during which the data taking has not yet started. Only 10 minutes
of the total passage through the SAA ($\sim 19$ minutes for a 550 km
equatorial orbit) were simulated, since the contribution of the isotopes
created at earlier times is, given the short lifetime, negligible. For the
long term activation, simulations were performed for a total of $\sim 72$
days, equivalent to the cumulative passage through the SAA in one year. A half
a orbit cool down was then added to the simulation in order to consider only
the long-lived isotopes contribution.
Simulations were made for two different altitudes (550 and 600 km) and several
orbit inclinations
($0^{\circ},1^{\circ},2^{\circ},3^{\circ},4^{\circ},5^{\circ},10^{\circ},15^{\circ}$).
Tab. 2 collects the most abundant isotopes created by the passage of the
particles inside the different parts of the satellite.

###  4.2 Analysis: count rates

After the simulation, the data were analyzed in order to retrieve the number
of events erroneously tagged as gamma-ray, either in the Compton or pair
regime. The rate of the falsely reconstructed pair events is extremely low
($<0.1$ counts/s). This is mainly due to the low energy nature of the
isotopes’ decay.

Figure 15: Analysis of the short-term activation caused by the passage of the
SAA at 550, considering different cool down times and orbit inclinations. For
the same inclination and cool down time, the two markers corresponds to the
total number of events (squares) and the number of events reconstructed as
Compton events (circles).

The total rate of decays of short-lived isotopes and the rate of the Compton
reconstructed events related to the passage through the SAA are shown in Fig.
15. Small changes are visible for orbits with inclinations smaller than $\sim
5^{\circ}$, with a $<1.5$ ratio between the results at $0^{\circ}$ and
$5^{\circ}$ inclination. A more than a factor of two increase is instead
observed in the reconstructed events at 10∘ and 15∘. The increase is
consistent with the raise in the observed flux shown in Fig. 13. As expected,
by considering a non-zero cool down time, the remaining events decreases
following an exponential trend.

Figure 16: Analysis of the activation from short-lived isotopes caused by the
passage of the SAA at 550 km (circles) and 600 km (triangles), considering
different cool down times and orbit inclinations. The different cool down
times are shown in different colors, as in Fig. 15. Top: Number of events
reconstructed as Compton events. Center: Ratio of the values obtained at the
two altitudes. Bottom: Ratio with respect to the perfectly equatorial orbit.

The rates increase with the altitude, as shown in Fig. 16, with a factor
between 1.5 and 2.5, depending on the inclination of the orbit $i$. The rates
increase with the inclination of the orbit $i$, with almost constant results
up to $i\sim 3^{\circ}$, gradually rising to a factor of 2 at $5^{\circ}$, a
factor of 5 at $5^{\circ}$, and reaching an order of magnitude difference for
an inclination of $15^{\circ}$. These changes do not depend on the trigger
choice. Either implementing a strict veto on the anticoincidence, i.e. without
using its segmentation but vetoing all the events with a signal in any of the
panel, or accepting all reconstructed events, the ratio remains constant and
the most favourable orbit is always a 550 km orbit with $i\lesssim 4^{\circ}$.
The change in the applied veto affects only the absolute number of
reconstructed events, with a $\sim$25% difference between the two mentioned
cases. It does not influence the rates variation at different inclination or
altitudes.

Figure 17: Comparison of the results of the activation caused by the general
on-orbit background and by the passage of the SAA, both from the short-lived
isotopes, without considering any cool down time, and from the long-live ones.
Results are shown for three different orbit inclinations. All the results were
calculated for an altitude of 550 km.

The rate of events reconstructed as Compton for the general background is
shown in Fig. 17, together with the results from one SAA passage without cool
down time, and its long-term activation after one year in orbit.
As shown in Fig. 12 (bottom panel), the flux of the secondary protons at
$i=10^{\circ}$ is significantly lower than that at $i=0^{\circ}$ for $E<5$
GeV. It only increases at higher energies, leading to a total factor of 3
decrease in the event rates at $10^{\circ}$. Between $10^{\circ}$ and
$15^{\circ}$, no changes in the secondary spectra are observed. The
atmospheric neutrons spectrum changes by only $\sim 20\%$, causing a $\sim
15\%$ increase in the rates. Changing from $i=0^{\circ}$ to $i=10^{\circ}$,
both the primary protons and alpha particles spectra change by more than a
factor of 2 for energies lower than the geomagnetic cutoff. This low energy
increase does not affect significantly the remaining counts, which change by
only $\sim 10\%$. Instead the SAA protons spectrum increases by a factor 2, at
low energies, to a factor of $\sim 7$, for energies higher than 10 MeV, which,
as previously shown, leads to a factor of 5 difference in the short-term
rates.
Up to an inclination of $10^{\circ}$, the main contributor to the total
activation is the decay of the isotopes created by the primary protons, the
secondary protons contribution being an order of magnitude smaller than that.
The counts from the alpha particles passage are one third of the ones related
to the primary protons, while the atmospheric neutrons rates are negligible.
The total SAA contribution, without considering any cool down time, can be
considered negligible for low inclinations ($i<5^{\circ}$), but it becomes
more important than the alpha particles contribution for $i\geqslant
10^{\circ}$, and the main contributor at $15^{\circ}$. For $i\geqslant
10^{\circ}$ it is necessary to consider some cool down time for the activation
rate from the trapped protons to become negligible. Instead, for low
inclination orbit, the short-term SAA activation contribution, comparable to
the one from the secondary protons, is always smaller than the contribution of
both the primary protons and alpha particles. The long-term SAA activation
contribution is higher than the short-term by a factor $\sim 1.5-1.6$ in the
considered inclination range. The SAA is the main contributor to the
background rates for high-inclination orbits, surpassing the rates from the
primary protons by a factor of 1.9.

Figure 18: Spectra of the reconstructed background Compton events for an orbit
inclination of 0∘ (top) or 15∘ (bottom). The total is calculated as the sum of
all the component from the activation except the SAA contribution. The SAA
contribution is divided in short-term, with a 0 s or 120 s cool down, and
long-term (SAA 1 yr).

###  4.3 Analysis: spectra

The spectra of the reconstructed Compton events with and without a recoil
electron track, from the activation simulations and from the extragalactic and
albedo photons along an orbit inclined by 0∘ and 15∘ are shown in Fig. 18. A
strict veto is used for both the side and top anticoincidence but no quality
cuts are applied to the reconstructed events.
The albedo and extragalactic photons are the main components of the background
until $\sim 400$ keV, afterward passed by the primary protons until $\sim~{}3$
MeV. The extragalactic photons contribution, while being the main one below
100 keV, it quickly loses importance at higher energies.
While the total rate of the events related to the passage through the SAA is
negligible for an equatorial orbit, they must be considered in the estimation
of the background for the 511 keV line observation. The contribution to this
line background from the short-term activation in an equatorial orbit
represents a fifth of the summed contributions of the other components without
cool down, and a tenth after two minutes. At an inclination of $15^{\circ}$
the short-term SAA contribution is not only dominant when no cool down is
considered, five times the sum of the other components, but it is still
comparable to the total after a two minutes cool-down time. The long-term
activation as well contributes to the 511 keV line background, similarly to
the short-term contribution when considering a two minutes cool down. For an
inclination of 15∘, the long-term contribution to the background is higher
than the sum of all the other components in the $\sim 150$ keV to $\sim 720$
keV energy range. The higher inclination orbits are therefore disfavoured.
The two most prominent lines that appear in the spectra are the 511 keV and
700 keV lines. The 511 keV is created by positrons annihilation. Positrons are
the results of the $\beta^{+}$ decay of isotopes such as 11C or 15O. The 700
keV line is visible in all activation spectra except in the SAA short-term
ones. It is mostly generated by the electron-capture decay of two long-lived
isotopes abundantly created in the CsI(Tl) crystals of the calorimeter: 132Cs
and 126I (see Table 2). The first radioisotope produces nuclear gamma rays of
668 keV at the same time as K$\alpha$ X-rays of 30 keV. The second one
produces 666 keV gamma rays together with 27 keV X-rays.

##  5 Conclusions

Thanks to the shielding provided by the geomagnetic field, Low Earth Orbits
(LEOs) present the lowest possible background for a gamma-ray space mission.
Satellites on LEO are nonetheless affected by a particle flux which depends on
both the altitude and the inclination of the orbit. This background has been
modeled starting from data of present and past experiments, considering the
possibility of small changes in the orbit altitude and inclination. The model
was later used to estimate the count rate from the decay of isotopes created
by the passage of these particles inside a possible future gamma-ray mission,
e-ASTROGAM (De Angelis et al (2017)), using the tool MEGAlib (Zoglauer et al
(2006)). This event rate was compared to the one from the residual activity
after a passage through the South Atlantic Anomaly (SAA), and its cumulative
effect after one year. The spectrum of the SAA has been computed using the
AE9/AP9-IRENE (Ginet et al (2013)) model.
The decay of the isotopes created by the primary cosmic-ray protons is found
to be the main contributor of the activation, the rates from alpha particles
being a fourth of that, while the contributions of the neutrons and secondary
protons can be considered negligible. The SAA contribution gains importance
when increasing the inclination of the orbit: from being negligible in nearly
equatorial orbit, it becomes the second contributor for a $10^{\circ}$
inclination, surpassing the event rate from the alpha particles interaction,
and the main contributor at $15^{\circ}$.
While the total event rate might be negligible, the SAA background for the 511
keV line observation must be always taken in consideration. While at low
inclination the short-term contribution to the line becomes quickly
negligible, at high inclination a long cool down is needed for the rates to
drop. For a $15^{\circ}$ inclination, the long-term SAA contribution is also
higher than the sum of all the other components in the 150 to 720 keV energy
range, while it is comparable to the alpha particles contribution for an
equatorial orbit.
A lower background and activation rate is found for an altitude of 550 km than
for 600 km, regardless of the inclination. The changes for a 50 km increase in
altitude are small in the case of the general background, but a factor between
two and three in the SAA case.
The best LEO for a gamma-ray mission is therefore found to be a low
inclination ($i<5$) and low altitude (550 km) orbit. Although, it should be
considered that a study on the atmospheric drag at the different altitudes,
while important in the planning of any LEO mission, is beyond the scope of
this paper. This kind of study might change the preference for a lower
altitude orbit but should not affect the obtained result on its inclination.

###### Acknowledgements.

This work has been carried out in the framework of the H2020 project AHEAD,
funded by the European Union.

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#  On the Metric Dimension
of Signed Graphs

Shahul Hameed K  Department of Mathematics, K M M Government Women’s College,
Kannur - 670004, Kerala, India. E-mail<EMAIL_ADDRESS>Remna K P
Department of Mathematics, Government Brennen College, Kannur - 670106,
Kerala, India.  Divya T  Department of Mathematics, Government Brennen
College, Kannur - 670106, Kerala, India.  Biju K  Department of Mathematics, P
R N S S College, Mattannur, Kannur - 670702, Kerala, India.
Rajeevan P  Department of Mathematics, Government Brennen College, Kannur -
670106, Kerala, India.  Santhosh G O  Department of Mathematics, Government
Brennen College, Kannur - 670106, Kerala, India.  Ramakrishnan K O  Department
of Mathematics, K M M Government Women’s College, Kannur - 670004, Kerala,
India. E-mail<EMAIL_ADDRESS>
###### Abstract

A signed graph $\Sigma$ is a pair $(G,\sigma)$, where $G=(V,E)$ is the
underlying graph in which each edge is assigned $+1$ or $-1$ by the signature
function $\sigma:E\rightarrow\\{-1,+1\\}$. In this paper, we extend the
extensively applied concepts of metric dimension and resolving sets for
unsigned graphs to signed graphs. We analyze the metric dimension of some well
known classes of signed graphs including a special case of signed trees. Among
other things, we establish that the metric dimension of a signed graph is
invariant under negation.

Key Words: Signed graph, Signed distance, Distance compatibility, Resolving
Set, Metric Dimension.

Mathematics Subject Classification (2010): 05C22, 05C50.

##  1 Introduction

A signed graph $\Sigma=(G,\sigma)$ is a graph $G=(V,E)$ together with a
signature function $\sigma:E\rightarrow\\{1,-1\\}$. In this paper, we extend
the definition of resolving sets and metric dimension of unsigned graphs to
signed graphs and compute metric dimensions of some classes of graphs and a
special class of signed trees. The signed distance concept for signed graphs
introduced in [9] by Shahul Hameed K et al., will be utilized for this
purpose.

All graphs in this paper are simple, finite and connected. The distance
between two vertices $u$ and $v$ in the underlying graph $G$ will be usually
denoted by $d(u,v)$ which is defined as the length of a shortest path between
them. If $\Sigma=(G,\sigma)$ is a signed graph, we denote the sign of a path
$P$ in $\Sigma$ by $\sigma(P)=\prod_{e\in P}\sigma(e)$. One of the shortest
paths from a vertex $u$ to a vertex $v$ is denoted by $P(u,v)$ and collection
of all such paths is denoted by $\mathcal{P}(u,v)$. There may be more than one
paths between two vertices in a non-geodetic graph and as such it is defined
[9] $\sigma_{\max}(uv)$ as $-1$ if $\sigma(P(u,v))=-1\ \forall\
P(u,v)\in\mathcal{P}(u,v)$ and $+1$, otherwise. Similarly $\sigma_{\min}(uv)$
as $+1$ if $\sigma(P(u,v))=+1\ \forall\ P(u,v)\in\mathcal{P}(u,v)$ and $-1$,
otherwise. A signed graph is said to be homogeneous if all the edges have the
same sign. Otherwise it is called non-homogeneous.

Given a connected signed graph $\Sigma=(G,\sigma)$ with $G=(V,E)$, there are
two types of signed distances $d^{\max}$ and $d^{\min}$ [9], defined by (i)
$d^{\max}(u,v)=\sigma_{\max}(uv)d(u,v)$ and (ii)
$d^{\min}(u,v)=\sigma_{\min}(uv)d(u,v)$ for all $u,v\in V$. Two vertices $u$
and $v$ in a connected signed graph $\Sigma$ are said to be (distance)
compatible [9], if $d^{\max}(u,v)=d^{\min}(u,v)$. $\Sigma$ itself is said to
be distance compatible or simply compatible [9], if every pair of its vertices
is distance compatible. In the case of a compatible signed graph $\Sigma$, we
denote the common value of $d^{\max}(u,v)$ and $d^{\min}(u,v)$ simply by
$d_{\Sigma}(u,v)$ or, if there is no scope for confusion, by $d(u,v)$, as in
the case of ordinary graphs. Also, in the case of a connected compatible
signed graph $\Sigma$, we denote by $\sigma(uv)$, not only the edge sign but
also the common value of $\sigma_{\max}$ and $\sigma_{\min}$.

The sign of a cycle $C$, denoted by $\sigma(C)$, in a signed graph
$\Sigma=(G,\sigma)$ is the product of its edge signs. i.e.,
$\sigma(C)=\prod_{e\in E(C)}\sigma(e)$. A cycle $C$ in a signed graph is said
to be positive if $\sigma(C)=1$. A signed graph is defined to be balanced if
every cycle in it is positive. Given a signed graph $\Sigma=(G,\sigma)$, its
negation is the signed graph $-\Sigma=(G,-\sigma)$. A signed graph $\Sigma$ is
said to be anti-balanced if its negation $-\Sigma$ is balanced. Geodetic
graphs are those graphs in which every pair of vertices will be joined by a
unique shortest path. Balanced signed graphs, anti-balanced signed graphs and
geodetic signed graphs are all examples of compatible signed graphs [9]. A
characterization of compatible signed graphs is given in [11], as follows.

######  Theorem 1.1 ([11]).

A $2$-connected, non-geodetic signed graph is compatible if and only if it has
no even negatively signed cycle $C_{2k}$ in which there are two diametrically
opposite vertices with distance $k$.

Resolving sets and basis for unsigned graphs were first introduced in the
$1970$s by Slater [7] and independently by Harary and Melter [3]. There are
many physical applications for these concepts on which extensive research is
still on. The reader may refer to [1, 2, 3, 5, 6, 7, 8, 10] for more details.
In order to extend these concepts to the case of distance compatible signed
graphs, first of all, let us have the following definitions.

Given a compatible signed graph $\Sigma=(V,E,\sigma)$, with respect to an
ordered subset $W=\\{w_{1},w_{2},\cdots w_{k}\\}\subset V$of cardinality $k$,
every vertex of $\Sigma$ has a $k$-vector representation as
$r_{\Sigma}(v|W)=(d_{\Sigma}(v,w_{1}),d_{\Sigma}(v,w_{2}),\cdots
d_{\Sigma}(v,w_{k}))$ called the metric representation of the vertex $v$. If
there is no scope for confusion, $r_{\Sigma}(v|W)$ will be simply denoted by
$r(v|W)$. An ordered subset $W$ of the vertex set $V$ of a compatible signed
graph $\Sigma=(V,E,\sigma)$ is called a resolving set if $r(v_{i}|W)\neq
r(v_{j}|W)$ for all vertices $v_{i}\neq v_{j}$ in $V$. A resolving set $W$ of
a compatible signed graph $\Sigma$ is called its basis if $W$ is a minimum
resolving set. The cardinality of a basis is called the metric dimension of
that signed graph, denoted by $\dim(\Sigma)$.

Regarding the organization of the paper, Section 2 deals with some basic but
important theorems regarding the resolving sets and the metric dimension of
signed graphs including the metric dimension of such signed graphs like signed
paths, signed cycles and signed stars. Section 3 and Section 4, respectively,
dwell upon exclusively with the metric dimension of signed wheels and signed
trees. Unless otherwise mentioned all the signed graphs under consideration
are distance compatible.

##  2 Main Results

######  Theorem 2.1.

Every resolving set of the underlying graph $G$ is a resolving set for the
signed graph $\Sigma=(G,\sigma)$.

###### Proof.

Let $V=\\{v_{1},v_{2},\dots,v_{n}\\}$ be the vertex set of $G$ and
$W=\\{v_{i_{1}},v_{i_{2}},\dots,v_{i_{k}}\\}$ be the resolving set of $G$.
Then $r(v_{p}|W)\neq r(v_{q}|W)\ \ \forall\ v_{p}\neq v_{q}\in V$. If possible
assume that $W$ is not a resolving set of $\Sigma$. Then
$r_{\Sigma}(v_{s}|W)=r_{\Sigma}(v_{t}|W)$ for some $v_{s},v_{t}\in V$. i.e.,
$(d_{\Sigma}(v_{s},v_{i_{1}}),\dots,d_{\Sigma}(v_{s},v_{i_{k}}))=(d_{\Sigma}(v_{t},v_{i_{1}}),\dots,d_{\Sigma}(v_{t},v_{i_{k}}))$.
This gives, $d_{\Sigma}(v_{s},v_{i_{x}})=d_{\Sigma}(v_{t},v_{i_{x}})\
{\text{for}}\ x=1,2,\dots,k$. Hence,
$|d_{\Sigma}(v_{s},v_{i_{x}})|=|d_{\Sigma}(v_{t},v_{i_{x}})|\ {\text{for}}\
x=1,2,\dots,k$. So, $d_{G}(v_{s},v_{i_{x}})=d_{G}(v_{t},v_{i_{x}})\
{\text{for}}\ x=1,2,\dots,k$ making $r_{G}(v_{s}|W)=r_{G}(v_{t}|W)\ \text{for
some}\ v_{s},v_{t}\in V(G)$, leading to a contradiction to the assumption.
Hence, $W$ is a resolving set for the signed graph $\Sigma$ also. ∎

######  Theorem 2.2.

For a signed graph $\Sigma=(G,\sigma)$, $1\leq\dim(\Sigma)\leq\dim(G)\leq
n-1$.

###### Proof.

The proof follows directly from the definitions since the co-ordinates for the
metric representation of a vertex can include negative integers also in the
case of a signed graph whereas that of the underlying graph allow only
positive integers. ∎

In view of the above theorem, we define the metric dimensional difference of a
signed graph $\Sigma=(G,\sigma)$, denoted by $\operatorname{mdd}(\Sigma)$, as
$\operatorname{mdd}(\Sigma)=\dim(G)-\dim(\Sigma)$. Thus
$\operatorname{mdd}(\Sigma)\geq 0$. The following theorem says that negating a
signed graph does not change the dimension.

######  Theorem 2.3.

If $\Sigma$ is a compatible signed graph, then $\dim(\Sigma)=\dim(-\Sigma)$

###### Proof.

Let $\Sigma=(G,\sigma)$ be the given signed graph and let
$-\Sigma=(G,\sigma^{\prime})$ be its negation where $\sigma^{\prime}=-\sigma$.
The sign of a shortest path $P(u,v)$ has the following relation under these
two signatures.

$\sigma^{\prime}(P(u,v))=(-1)^{d(u,v)}\sigma(P(u,v))$ (2.1)

Let $\dim(\Sigma)=k$ and $W=\\{w_{1},w_{2},\cdots,w_{k}\\}$ be a basis for
$\Sigma$. We claim that this $W$ is a basis for $-\Sigma$ also. To prove this,
first of all we will establish that $W$ is a resolving set for $-\Sigma$. As
$W$ is a resolving set for $\Sigma$, given $u,v\in V(\Sigma)$, there exists
$i$ such that $1\leq i\leq k$ with $d_{\Sigma}(u,w_{i})\neq
d_{\Sigma}(v,w_{i})$. As our motive is to prove that

$d_{-\Sigma}(u,w_{i})\neq d_{-\Sigma}(v,w_{i})$ (2.2)

for the same $i$ we proceed with the following three cases.
Case(i): Both the shortest paths $P(u,w_{i})$ and $P(v,w_{i})$ are of even
length. In this case, using (2.1),
$\sigma^{\prime}(P(u,w_{i}))=\sigma(P(u,w_{i}))$ and
$\sigma^{\prime}(P(v,w_{i}))=\sigma(P(v,w_{i}))$ so that
$d_{-\Sigma}(u,w_{i})=d_{\Sigma}(u,w_{i})$ and
$d_{-\Sigma}(v,w_{i})=d_{\Sigma}(v,w_{i})$ which proves the requisite
condition in (2.2).
Case(ii): Both the shortest paths $P(u,w_{i})$ and $P(v,w_{i})$ are of odd
length. Here, using (2.1), $\sigma^{\prime}(P(u,w_{i}))=-\sigma(P(u,w_{i}))$
and $\sigma^{\prime}(P(v,w_{i}))=-\sigma(P(v,w_{i}))$ so that
$d_{-\Sigma}(u,w_{i})=-d_{\Sigma}(u,w_{i})$ and
$d_{-\Sigma}(v,w_{i})=-d_{\Sigma}(v,w_{i})$ which proves the condition in
(2.2).
Case(iii): The shortest path $P(u,w_{i})$ is of odd length and $P(v,w_{i})$ is
of even length. Then it is immediate that $d_{G}(u,w_{i})\neq d_{G}(v,w_{i})$
and as such (2.2) remains true. Thus in all cases, the condition in (2.2)
holds good which proves the claim that $W$ is a resolving set for $-\Sigma$
also. To prove that it is a basis, if possible let $W^{\prime}$ be a resolving
set for $-\Sigma$ with $|W^{\prime}|<|W|$, then it leads to a contradiction
since $W^{\prime}$ will then be a resolving set for $-(-\Sigma)=\Sigma$. ∎

Denoting an all-positive signed complete graph by $K_{n}^{+}$ and an all-
negative signed complete graph by $K_{n}^{-}$, we have the following.

######  Corollary 2.4.

$\dim(K_{n}^{+})=\dim(K_{n}^{-})=n-1$.

###### Proof.

Note that $K_{n}^{-}=-K_{n}^{+}$ and apply Theorem 2.3. ∎

######  Corollary 2.5.

The metric dimension of an all-negative signed cycle is $2$.

###### Proof.

If $C_{n}^{\sigma}=(C_{n},\ \sigma)$ is the all negative signed cycle, then
$-C_{n}^{\sigma}=C_{n}$ and the well known result [2] $\dim(C_{n})=2$
completes the proof from Theorem 2.3. ∎

In the case of an unsigned complete graph $K_{n}$, it is well known that
$\dim(K_{n})=n-1$. It is not generally true in the case of signed complete
graphs as illustrated in the following example, but we have the bounds for the
metric dimension of signed complete graphs given in Theorem 2.7.

######  Example 2.6.

Consider $K_{3}^{\sigma_{1}}$ where there is only one negative edge. Then
using Theorem 2.9, $\dim(K_{3}^{\sigma_{1}})=1$. Also for a complete signed
graph with the underlying graph $K_{4}$ having only one negative edge, its
metric dimension can be easily verified to be $2$. Moreover using the possible
vectors $(0,\pm 1),\ (\pm 1,0),\ (1,1),\ (-1,1),\ (1,-1),\ (-1,-1)$, one can
really construct signed complete graphs of order up to $6$ with the metric
dimension $2$.

From the above discussion, we arrive at the following theorem in the case of
signed complete graphs of order greater than $3$.

######  Theorem 2.7.

The metric dimension of a signed complete graph
$K_{n}^{\sigma}=(K_{n},\sigma)$ satisfies $2\leq\dim(K_{n}^{\sigma})\leq n-1$
for all $n\geq 4$.

######  Theorem 2.8.

The metric dimension of a signed path is $1$.

###### Proof.

Let $P_{n}^{\sigma}=(P_{n},\sigma)$ be the given signed path where
$P_{n}:u_{1}u_{2}\cdots u_{n}$. $P_{n}^{\sigma}$ being a compatible signed
graph, letting $W=\\{u_{1}\\}$, it can be easily seen that
$r(u_{i}|W)=(\sigma(u_{1}u_{i})(i-1))$ for $i\neq 1$ and $(0)$ for $i=1$. This
shows that $W$ is a resolving set for $P_{n}^{\sigma}$ and
$\dim(P_{n}^{\sigma})=1$.
Indeed, there is a shorter proof, if we use Theorem 2.2 as follows.
$1\leq\dim(P_{n}^{\sigma})\leq\dim(P_{n})=1$. ∎

For an unsigned graph, the converse is also true and an unsigned path is
characterized to be the only graph having metric dimension $1$. But in the
case of signed graphs, it is not the case as proved in the following theorems.
Note that in the case of an unsigned cycle $C_{n},\ \dim(C_{n})=2$. We denote
by $d^{+}(v)$, the number of positive edges incident with the vertex $v$ in a
signed graph and similarly $d^{-}(v)$ denotes the number of negative edges
incident with $v$. The net-degree of a vertex $v$ in a signed graph is
$d^{\pm}(v)=d^{+}(v)-d^{-}(v)$. Also, open $k$-neighbourhood of a vertex $v$
in a signed graph $\Sigma=(G,\sigma)$, denoted by $N_{k}(v)$, is
$N_{k}(v)=\\{u\in V(\Sigma):d(u,v)=\pm\ k\\}$.

######  Theorem 2.9.

Let $C_{n}^{\sigma}=(C_{n},\sigma)$ be a signed cycle. Then
$\dim(C_{n}^{\sigma})=1$ if and only if there exists a vertex $u$ with
$d^{\pm}(u)=0$ and for $v,w\in N_{k}(u),\ \sigma(uv)\neq\sigma(uw)$ whenever
$v\neq w$ for all $k=1,2,\cdots,\lfloor n/2\rfloor$.

###### Proof.

First we deal with the case when $n=2p+1$. The case for even cycle follow
along the same lines. Let the underlying odd cycle $C_{2p+1}$ has the vertex
set $V$ and let $u_{1}\in V$ be the vertex with $d^{\pm}(u_{1})=0$. We claim
that $W=\\{u_{1}\\}$ is a resolving set and hence a basis, if for $v,w\in
N_{k}(u_{1}),\ \sigma(u_{1}v)\neq\sigma(u_{1}w)$ whenever $v\neq w$ for all
$k=1,2,\cdots,p$. Clearly, if $v\in N_{i}(u_{1})$ and $w\in N_{j}(u_{1})$ for
$i\neq j$, then $r(v|W)\neq r(w|W)$ as $d(v,u_{1})\neq d(w,u_{1})$. Therefore,
we choose $v,w\in N_{k}(u_{1})$ with $v\neq w$ where $k=1,2,\cdots,p$. Then by
the assumption that $\sigma(u_{1}v)\neq\sigma(u_{1}w)$, $d(v,u_{1})\neq
d(w,u_{1})$ proving that $r(v|W)\neq r(w|W)$. This establishes one way
implication. For the converse, first we note that existence of a vertex
$u_{1}$ with net degree zero is essential and only such vertices can come in a
basis of cardinality $1$. Moreover, if for some $v,w\in N_{k}(u_{1}),\
\sigma(u_{1}v)=\sigma(u_{1}w)$, then $r(v|W)=r(w|W)$ where $W=\\{u_{1}\\}$,
implying that, it is not a resolving set and hence not a basis.

In the case of an even signed cycle, using Theorem 1.1, it is compatible only
when it is balanced. Hence to check its metric dimension is $1$, in view of
Corollary 2.5, it is first of all necessary that there must exist a vertex of
net degree $0$ and thereafter the remaining assumptions in the theorem. ∎

Now we deal with the metric dimension of signed stars. In the case of all-
positive and all-negative signed stars, the dimension, with the help of the
established result [2] and Theorem 2.3, is $n-1$.

######  Theorem 2.10.

$\dim(K_{1,n}^{\sigma})=n-2$ for $n\geq 3$ when it is non-homogeneous.

###### Proof.

Let $V=\\{u\\}\cup V_{1}\cup V_{2}$ be the vertex set where $u$ is the central
vertex of $K_{1,n}$ and $V_{1}=\\{v_{1},v_{2},\cdots v_{l}\\}$ and
$V_{2}=\\{u_{1},u_{2},\cdots u_{m}\\}$ be the vertices of net-degrees $1$ and
$-1$ respectively with $l+m=n$. Since $K_{1,n}^{\sigma}$ is non-homogeneous,
$l\geq 1$ and $m\geq 1$. Define an ordered set
$W=\\{v_{1},v_{2},...v_{l-1},u_{1},u_{2},...u_{m-1}\\}$. We claim that this is
the required basis for $K_{1,n}^{\sigma}$. First we prove that $W$ is a
resolving set, for which it is enough to establish that metric representation
of $u,v_{l}$ and $u_{m}$ are different. Denoting the $i$-th co-ordinate of
$r(u|W)$ by $r(u|W)(i)$,

$\displaystyle r(u|W)(i)$ $\displaystyle=1\mbox{\ for\ }1\leq i\leq l-1$
$\displaystyle=-1\mbox{\ for\ }l\leq i\leq n-2$

and

$\displaystyle r(v_{l}|W)(i)$ $\displaystyle=2\mbox{\ for\ }1\leq i\leq l-1$
$\displaystyle=-2\mbox{\ for\ }l\leq i\leq n-2$

Similarly,

$\displaystyle r(u_{m}|W)(i)$ $\displaystyle=-2\mbox{\ for\ }1\leq i\leq l-1$
$\displaystyle=2\mbox{\ for\ }l\leq i\leq n-2$

Thus all the vertices have different metric representations with respect to
$W$ proving that it is a resolving set. To show that $W$ is minimum, if
possible let $W^{\prime}$ be a resolving set with $|W^{\prime}|<|W|$. Assume
$|W^{\prime}|=k<n-2$ and let $W^{\prime}=\\{x_{1},x_{2},\cdots
x_{h}\\}\cup\\{x_{h+1},x_{h+2},\cdots,x_{k}\\}=W_{1}\cup W_{2}$, say, such
that vertices in $W_{1}$ are of net-degree $1$ and $W_{2}$ contains vertices
of net-degree $-1$ and possibly, they may include the central vertex of the
signed star too. Then, outside $W_{1}\cup W_{2}$ there exist at least four
vertices from among them we can pick at least two vertices $x_{i}$ and $x_{j}$
such that $\sigma(ux_{i})\sigma(ux_{j})=1$, which implies that
$r(x_{i}|W^{\prime})=r(x_{j}|W^{\prime})$, a contradiction to the fact that
$W^{\prime}$ is a resolving set. Hence the result. ∎

##  3 Metric dimension of signed wheels

This section deals exclusively with the metric dimension of signed wheels and
allied results on resolving sets for a signed wheel. A wheel $W_{n}$ is the
join of a cycle $C_{n}$ and $K_{1}$. The vertex of $K_{1}$ is called the
central vertex of the wheel $W_{n}$ and the vertices on the cycle is often
referred to as its rim vertices. The following theorem, found in [10], gives
the metric dimension of an unsigned wheel.

######  Theorem 3.1 ([10]).

$\dim(W_{n})=\left\\{\begin{array}[]{rlll}\Bigl{\lfloor}\dfrac{2n+2}{5}\Bigr{\rfloor},&\text{if
}n\geq 7,\\\ 2,&\text{if }n=4,5,\\\ 3,&\text{if }n=3,6.\end{array}\right.$
(3.1)

We use the notation $W_{n}^{\sigma}=(W_{n},\sigma)$ to denote a signed wheel.
For any two vertices $u$ and $v$ in $W_{n}$, either $d(u,v)=1$ or $d(u,v)=2$
as the diameter of any wheel is at most $2$. So to discuss the distance
compatibility of $W_{n}^{\sigma}$, where $n\geq 4$, in view of Theorem 1.1, it
is enough to consider only $C_{4}$ with one vertex as the central vertex of
$W_{n}^{\sigma}$. We establish the criteria for the distance compatibility of
signed wheels as follows in which $C_{4}^{-}$ denotes a negative cycle with
four vertices.

######  Theorem 3.2.

A signed wheel $W_{n}^{\sigma}$, where $n\geq 4$, is distance compatible if
and only if it is $C_{4}^{-}$-free where one of the vertices of the cycle is
the central vertex of $W_{n}^{\sigma}$.

###### Proof.

Consider a distance compatible signed wheel $W_{n}^{\sigma}$ with vertex set
$\\{v,v_{1},v_{2},....v_{n}\\}$, where $v$ is the central vertex of
$W_{n}^{\sigma}$. First of all, assume that $W_{n}^{\sigma}$ has a negative
even cycle $C_{4}^{-}$ with $C_{4}:vv_{1}v_{2}v_{3}v$. Then $C_{4}$ has either
one negative edge or three negative edges. In both the cases
$\sigma_{max}(v_{1}v_{3})=1$ and $\sigma_{min}(v_{1}v_{3})=-1$ and hence
$d^{max}(v_{1},v_{3})\neq d^{min}(v_{1},v_{3})$, which shows $W_{n}^{\sigma}$
is not distance compatible, a contradiction.
For the converse, assume that $W_{n}^{\sigma}$ has no negative even cycle
$C_{4}$ with one of the vertices on it as the central vertex of the wheel. Let
$u$ and $v$ be any two vertices of $W_{n}$. Then the following two cases
arise.
$\boldsymbol{Case$$(i)}$: $u$ and $v$ are adjacent vertices.
If $\sigma(uv)=1$, then $\sigma_{max}(uv)=1=\sigma_{min}(uv)$ which makes
$d^{max}(u,v)=d^{min}(u,v)=1$.
If $\sigma(uv)=-1$, then $\sigma_{max}(uv)=-1=\sigma_{min}(uv)$ implying that
$d^{max}(u,v)=d^{min}(u,v)=-1$. So $u$ and $v$ are distance compatible.
$\boldsymbol{Case$$(ii)}$: $u$ and $v$ are nonadjacent vertices so that
$d(u,v)=2$. Assume that $\sigma_{max}(uv)\neq\sigma_{min}(uv)$. Since the case
$\sigma_{max}(uv)=-1$ and $\sigma_{min}(uv)=1$ is not possible simultaneously,
we have $\sigma_{max}(uv)=1$ and $\sigma_{min}(uv)=-1$. So by definition,
there exist a positive and negative $u-v$ paths of length $2$. These two paths
together makes a negative even cycle $C_{4}^{-}$ with one vertex as the
central vertex of the wheel, a contradiction. Hence
$\sigma_{max}(uv)=\sigma_{min}(uv)$ which implies $d^{max}(u,v)=d^{min}(u,v)$
and so $u$ and $v$ are distance compatible. ∎

######  Lemma 3.3.

A set of cardinality one cannot resolve a signed wheel $W_{n}^{\sigma}\ \
\forall\ n\geq 3$. In other words, $\dim(W_{n}^{\sigma})\geq 2$.

###### Proof.

$\boldsymbol{Case$$(1)}$: Let $B=\\{v_{0}\\}$, where $v_{0}$ is the central
vertex. Then there exists three rim vertices $v_{1},v_{2},v_{3}$ with
$r(v_{i}|B)=1$. Since $\sigma(v_{0}v_{1})=\sigma(v_{0}v_{2})$ or
$\sigma(v_{0}v_{1})=\sigma(v_{0}v_{3})$, this implies $\
r_{\Sigma}(v_{1}|B)=r_{\Sigma}(v_{2}|B)$ or
$r_{\Sigma}(v_{1}|B)=r_{\Sigma}(v_{3}|B)$. Hence, $B=\\{v_{0}\\}$ does not
resolve $W_{n}^{\sigma}$.
$\boldsymbol{Case$$(2)}$: Assume that $B\neq\\{v_{0}\\}$ and let
$B=\\{v_{i}\\},v_{i}\in V\backslash\\{v_{0}\\}$. Then
$\sigma(v_{i-1}v_{i})=\sigma(v_{i+1}v_{i})$ or
$\sigma(v_{i-1}v_{i})=\sigma(v_{0}v_{i})$. Therefore, $\
r_{\Sigma}(v_{i-1}|B)=r_{\Sigma}(v_{i+1}|B)$ or
$r_{\Sigma}(v_{i-1}|B)=r_{\Sigma}(v_{0}|B)$ showing that $B$ cannot resolve
$W_{n}^{\sigma}$. Hence the proof. ∎

######  Remark 3.4.

For a wheel $W_{n},n>6$, let $\dim(W_{n})=k$. To show that the central vertex
of this unsigned wheel should not be a member of any of its basis, one can
have the following argument. If we choose any set $B$ with $k-1$ rim vertices
then there are two rim vertices $v_{i},v_{j}$ such that
$r(v_{i}|B)=r(v_{j}|B)=(x_{1},x_{2},\cdots,x_{k-1})$, say. So if
$B_{1}=\\{v_{0}\\}\cup\ B$, where $v_{0}$ is the central vertex, then
$r(v_{i}|B_{1})=r(v_{j}|B_{1})=(1,x_{1},x_{2},\cdots,x_{k-1})$. So $B_{1}$
does not resolve $W_{n}$ when $n>6$. This completes the argument for the above
result. The same is the case with the signed wheel also as shown in the
following theorem.

######  Theorem 3.5.

For $n=3,4,5,6$, the central vertex may be chosen as an element of a basis of
$W_{n}^{\sigma}$ and for $n>6$, the central vertex should not be an element of
a basis of $W_{n}^{\sigma}$.

###### Proof.

For $n=3$, the underlying wheel (note that it is a complete graph $K_{4}$) has
a basis $B$ which includes its central vertex. So this basis $B$ resolves
$W_{n}^{\sigma}$ also and if we choose a set $B_{1}$ with two elements, which
includes the central vertex $v_{0}$ and a rim vertex, without loss of
generality, let it be $v_{1}$, then there exist two rim vertices with
$r(v_{2}|B_{1})\ =r(v_{3}|B_{1})\ =(1,1)$. So if
$\sigma(v_{0}v_{2})\neq\sigma(v_{0}v_{3})$, then $r_{\Sigma}(v_{2}|B_{1})\
\neq r_{\Sigma}(v_{3}|B_{1})$. By Lemma 3.3, as no singleton set resolves a
signed wheel, $B_{1}$ will act as a basis of $W_{3}^{\sigma}$. On the other
hand, if $\sigma(v_{0}v_{2})=\sigma(v_{0}v_{3})$, then $B$ will becomes a
basis of $W_{3}^{\sigma}$.
For $n=4$, $B_{1}=\\{v_{0},v_{1}\\}$, where $v_{0}$ is the central vertex and
$v_{1}$ is a rim vertex, does not resolve $W_{n}$, since we have
$r(v_{2}|B_{1})\ =r(v_{4}|B_{1})\ =(1,1)$. So the central vertex cannot be an
element of any basis of $W_{4}$. But if

$\sigma(v_{0}v_{2})\neq\sigma(v_{0}v_{4})$ (3.2)

, then $r_{\Sigma}(v_{2}|B_{1})\ \neq r_{\Sigma}(v_{4}|B_{1})$. By Lemma 3.3,
this gives $B_{1}$ as basis of $W_{4}^{\sigma}$. But when,
$\sigma(v_{0}v_{2})=\sigma(v_{0}v_{4})$, $B_{1}$ does not resolve
$W_{4}^{\sigma}$. So the central vertex cannot be an element of basis of
$W_{4}^{\sigma}$. Hence if Equation (3.2) holds, by Lemma 3.3,
$W_{4}^{\sigma}$ can have a basis with central vertex.
For $n=5$, $B_{1}=\\{v_{0},v_{1}\\}$, where $v_{0}$ is the central vertex and
$v_{1}$ is a rim vertex does not resolve $W_{5}$, since we have
$r(v_{2}|B_{1})\ =r(v_{5}|B_{1})\ =(1,1)$ and $r(v_{3}|B_{1})\
=r(v_{4}|B_{1})\ =(1,2)$. So the central vertex cannot be an element of basis
of $W_{5}$. Now, if

$\sigma(v_{0}v_{2})\neq\sigma(v_{0}v_{5})\ and\
\sigma(v_{0}v_{3})\neq\sigma(v_{0}v_{4})$ (3.3)

, then $r_{\Sigma}(v_{i}|B_{1})\ \neq r_{\Sigma}(v_{j}|B_{1})$, $\forall\
v_{i},v_{j}\in V$. Therefore, by Lemma 3.3, $B_{1}$ acts as a basis of
$W_{5}^{\sigma}$. Moreover, if $\sigma(v_{0}v_{2})=\sigma(v_{0}v_{5})$ or
$\sigma(v_{0}v_{3})=\sigma(v_{0}v_{4})$, then $B_{1}$ cannot resolve
$W_{5}^{\sigma}$. Hence if Equation (3.3) holds, by Lemma 3.3,
$W_{5}^{\sigma}$ can have a basis with the central vertex.
For $n=6$, the underlying graph has a basis $W$ which includes the central
vertex, which resolves $W_{6}^{\sigma}$ also and if we choose a set $B_{1}$
with two vertices, which includes $v_{0}$ and an arbitrary rim vertex $v_{1}$,
say, then there will be three vertices with same metric representation in the
underlying graph and let it be $v_{a},v_{b},v_{c}$. As such,

$\sigma(v_{0}v_{a})=\sigma(v_{0}v_{b})\ or\
\sigma(v_{0}v_{a})=\sigma(v_{0}v_{c}).$ (3.4)

Therefore, $r_{\Sigma}(v_{a}|B_{1})\ =r_{\Sigma}(v_{b}|B_{1})$ or
$r_{\Sigma}(v_{a}|B_{1})\ =r_{\Sigma}(v_{b}|B_{1})$. This shows that $B_{1}$
does not resolve $W_{n}^{\sigma}$. Now, a set with 3 rim vertices resolves
$W_{6}^{\sigma}$ and let $B_{1}=\\{v_{k1},v_{k2},v_{k3}\\}\subseteq
V\setminus\\{v_{0}\\}$ be a set with $r(v_{i}|B_{1})\neq\
r(v_{j}|B_{1});\forall\ v_{i},v_{j}\in V$. By Theorem 2.1, this gives $\
r_{\Sigma}(v_{i}|B_{1})\neq\ r_{\Sigma}(v_{j}|B_{1})\ \forall\ v_{i},v_{j}\in
V$. We proceed with the following cases and subcases.
Case - A Consider any set with two vertices without the central vertex, let it
be $B_{2}=\\{v_{1},v_{4}\\}\subseteq V$ or $B_{2}=\\{v_{1},v_{2}\\}\subseteq
V$ or $B_{2}=\\{v_{1},v_{6}\\}\subseteq V$. Then, $r(v_{p}|B_{2})=\
r(v_{q}|B_{2})$ for $v_{p},v_{q}\in V\setminus\\{v_{0}\\}$. Let
$S=\\{(v_{p},v_{q})|\sigma(v_{0}v_{p})=\sigma(v_{0}v_{q})\\}$.
Subcase-1: If $(v_{p},v_{q})\in S$ then $\
r_{\Sigma}(v_{p}|B_{2})=r_{\Sigma}(v_{q}|B_{2})$. Therefore, $B_{2}$ does not
resolve $W_{6}^{\sigma}$
Subcase-2: If all such $(v_{p},v_{q})\notin S$ then $\
r_{\Sigma}(v_{p}|B_{2})\neq r_{\Sigma}(v_{q}|B_{2})$. So, $B_{2}$ resolves
$W_{6}^{\sigma}$.
Case - B If $B_{2}=\\{v_{1},v_{3}\\}$ or $\\{v_{1},v_{5}\\}$ then
$r(v_{0}|B_{2})=\ r(v_{p}|B_{2})=(1,1)$ for a $v_{p}\in V\setminus\\{v_{0}\\}$
1) If $\ r_{\Sigma}(v_{0}|B_{2})=(1,1)$ and
$r_{\Sigma}(v_{p}|B_{2})\in\\{(1,-1),(-1,1),(-1,-1)\\};\ v_{p}\in V\setminus
B_{2}$ where $p\neq 0$, $B_{2}$ resolves $W_{6}^{\sigma}$.
2) If $\ r_{\Sigma}(v_{0}|B_{2})=(-1,-1)$ and
$r_{\Sigma}(v_{p}|B_{2})\in\\{(1,-1),(-1,1),(1,1)\\};\ v_{p}\in V\setminus
B_{2}$ where $p\neq 0$, $B_{2}$ resolves $W_{6}^{\sigma}$.
3) If $\ r_{\Sigma}(v_{0}|B_{2})=(1,-1)$ and
$r_{\Sigma}(v_{p}|B_{2})\in\\{(-1,1),(1,1),(-1,-1)\\};v_{p}\in V\setminus
B_{2}$ where $p\neq 0$, $B_{2}$ resolves $W_{6}^{\sigma}$.
4) If $\ r_{\Sigma}(v_{0}|B_{2})=(-1,1)$ and
$r_{\Sigma}(v_{p}|B_{2})\in\\{(1,1),(1,-1),(-1,-1)\\}\ ;v_{p}\in V\setminus
B_{2}$ where $p\neq 0$, $B_{2}$ resolves $W_{6}^{\sigma}$. Otherwise $B_{2}$
does not resolve $W_{6}^{\sigma}$.
By Lemma 3.3, no singleton set resolves the signed wheel and by Theorem 2.2,
$\dim(W_{6}^{\sigma})=2$ [By Subcase 2 and Case B]. If the Subcase 2 and Case
B do not hold then $\dim(W_{6}^{\sigma})=3$, so the central vertex can be an
element of a basis of $W_{6}^{\sigma}$.
For $n>6$, if $\dim(W_{n})=k$, then the underlying graph does not include the
central vertex in its basis. So if we take
$B=\\{v_{0},v_{m_{1}},v_{m_{2}},...v_{m_{k-1}}\\}$, then there exist
$(v_{p},v_{q})$ such that $r(v_{p}|B)\ =r(v_{q}|B)$, for some $v_{p},v_{q}\in
V$.
If $\sigma(v_{0}v_{p})\neq\sigma(v_{0}v_{q})$ for $(v_{p},v_{q})$ satisfying
$r(v_{p}|B)\ =r(v_{q}|B)$, then $r_{\Sigma}(v_{i}|B)\ \neq
r_{\Sigma}(v_{j}|B)$, $\forall\ v_{i},v_{j}\in V$ and as such, $B$ resolves
$W_{n}^{\sigma}$. But if we choose $B_{1}=B\setminus\\{v_{0}\\}$ in graph
satisfying the above condition that $\sigma(v_{0}v_{p})\neq\sigma(v_{0}v_{q})$
for $(v_{p},v_{q})$ where $r(v_{p}|B)\ =r(v_{q}|B)$, then $B_{1}$ resolves
$W_{n}^{\sigma}$. This shows that no basis of $W_{n}^{\sigma}$ can include the
central vertex for $n>6$. ∎

The following theorem gives the metric dimensional difference
$\operatorname{mdd}(W_{n}^{\sigma})$.

######  Theorem 3.6.

Let $W_{n}^{\sigma}=(W_{n},\sigma)$ be a compatible signed wheel where $n\geq
3$. Then $\operatorname{mdd}(W_{n}^{\sigma})\leq 1$.

###### Proof.

For simplicity of notations, we use $G$ for denoting the underlying wheel and
$\Sigma$ for the signed wheel and deal with the following cases depending on
the values of $n$.
Case-1: [ n=3 ]
Let V=$\\{v_{0},v_{1},v_{2},v_{3}\\}$ be the vertex set. Then $\dim(G)=3$ from
Theorem 3.1. By Theorem 2.1, there exist a set with $3$ vertices which
resolves $\Sigma$ also. Let $\\{v_{a},v_{b},v_{c}\\}$ $\subseteq V$ resolve
$\Sigma$. If we take any set with two vertices, let it be
$B=\\{v_{a},v_{b}\\}$ then easy computations give the following values as the
case may be.
$\ r(v_{c}|B)=r(v_{d}|B)=(1,1)$ where $v_{c},v_{d}\in V$. Therefore, if $\
r_{\Sigma}(v_{c}|B)=(1,1)$ and $\
r_{\Sigma}(v_{d}|B)\in\\{(1,-1),(-1,1),(-1,-1)\\}$ or $\
r_{\Sigma}(v_{c}|B)=(-1,-1)$ and $\
r_{\Sigma}(v_{d}|B)\in\\{(1,-1),(-1,1),(1,1)\\}$ or
$r_{\Sigma}(v_{c}|B)=(-1,1)$ and $\
r_{\Sigma}(v_{d}|B)\in\\{(1,1),(-1,-1),(1,-1)\\}$ or $\
r_{\Sigma}(v_{c}|B)=(1,-1)$ and $\
r_{\Sigma}(v_{d}|B)\in\\{(1,1),(-1,-1),(-1,1)\\}$. These different sets of
values provide

$r_{\Sigma}(v_{c}|B)\neq\ r_{\Sigma}(v_{d}|B)$ (3.5)

making $\\{v_{a},v_{b}\\}$ as a resolving set for $\Sigma$. By Lemma 3.3, as a
singleton set does not resolve $\Sigma$, $\\{v_{a},v_{b}\\}$ acts as basis. If
the above condition in (3.5) does not hold, then by Theorem 2.2
$\dim(\Sigma)=3$. This proves that $\dim(W_{3}^{\sigma})$ is either $2$ or
$3$.
Case-2: [$n=4,5$ ]
Here, $\dim(G)=2$. Therefore, there exist $B=\\{v_{p},v_{q}\\}\subseteq V$
such that $\ r(v_{i}|B)\neq\ r(v_{j}|B);\forall\ v_{i},v_{j}\in V$. By Theorem
2.1 and Theorem 3.5, this means $\ r_{\Sigma}(v_{i}|B)\neq\
r_{\Sigma}(v_{j}|B)\ \forall\ v_{i},v_{j}\in V$. Also by Lemma 3.3, singleton
set does not resolve $\Sigma$ gives $\dim(W_{n}^{\sigma})=2$ when $n=4,5$.
Case-3: [ $n=6$ ]
In this case, $\dim(G)=3$ and letting $V=\\{v_{0},v_{1},v_{2}....v_{6}\\}$,
there exists a basis $B_{1}=\\{v_{k1},v_{k2},v_{k3}\\}\subseteq V$ such that
$r(v_{i}|B_{1})\neq\ r(v_{j}|B_{1});$ for all $v_{i},v_{j}\in V$. So by
Theorem 2.1, $\ r_{\Sigma}(v_{i}|B_{1})\neq\ r_{\Sigma}(v_{j}|B_{1})\ \forall\
v_{i},v_{j}\in V$
If Subcase 2 of Case A and Case B of Theorem 3.5 holds, then $\dim(\Sigma)=2$.
Otherwise $\dim(\Sigma)=3$. This proves that, $\dim(W_{6}^{\sigma})$ is either
$2$ or $3$.
Case-4 [ $n>6$ ]
Let $V=\\{v_{0},v_{1},v_{2},...v_{n}\\}$ where $v_{0}$ is the central vertex
and $\\{v_{1},v_{2},...v_{n}\\}$ are the rim vertices. If $\dim(G)=k$, then
let $B_{1}=\\{v_{i1},v_{i2},...v_{ik}\\}\subseteq V\setminus\\{v_{0}\\}$
resolve $G$. By Theorem 2.1, $B_{1}$ resolves $\Sigma$ too. If we take any set
with $k-1$ vertices, $B_{2}=\\{v_{j1},v_{j2},...v_{jk-1}\\}\subseteq
V\setminus\\{v_{0}\\}$ then $r(v_{s}|B_{2})=\ r(v_{t}|B_{2})$ for some
$v_{s},v_{t}\in V$. Define $S=\\{(v_{s},v_{t})\|\ r(v_{s}|B_{2})=\
r(v_{t}|B_{2})\\}$ and partition S into two subsets $S_{1}$ and $S_{2}$ such
that $S_{1}=\\{(v_{s},v_{t})\in S|\sigma(v_{0}v_{s})=\sigma(v_{0}v_{t})\\}$
and $S_{2}=\\{(v_{s},v_{t})\in S|\sigma(v_{0}v_{s})\neq\sigma(v_{0}v_{t})\\}$.
Subcase-A
If at least one $(v_{s},v_{t})\in S$ belongs to $S_{1}$ then $\
r_{\Sigma}(v_{s}|B_{2})=r_{\Sigma}(v_{t}|B_{2})$. Therefore, $B_{2}$ does not
resolve $\Sigma$ and along the same line of argument, it can be shown that no
other set with cardinality less than $k-1$ resolves $\Sigma$. By theorem 2.2
this proves $\dim(\Sigma)=k$.
Subcase-B
If $S=S_{2}$, then $r_{\Sigma}(v_{s}|B_{2})\neq r_{\Sigma}(v_{t}|B_{2})$ $\
\forall(v_{s},v_{t})\in S$ provides $B_{2}$ to resolve $\Sigma$. To settle the
minimality, if we take any set of $k-2$ vertices,
$W_{3}=\\{v_{l1},v_{l2},...v_{lk-2}\\}\subseteq V\ \backslash\\{v_{0}\\}$ then
$r(v_{a}|B_{3})=r(v_{b}|B_{3})=\underbrace{(2,2....2)}_{(k-2)times}$ for some
$v_{a}$ and $v_{b}\in V$. Define $S_{3}=\\{(v_{a},v_{b})\in
V:r(v_{a}|B_{3})=r(v_{b}|B_{3})\\}$. Since $n>6,|S_{3}|>2$, and so there exist
at least three vertices $v_{a},v_{b},v_{c}\in V$ such that
$r(v_{a}|B_{3})=r(v_{b}|B_{3})=r(v_{c}|W_{3})=\underbrace{(2,2....2)}_{(k-2)times}$.
This gives, $\sigma(v_{0}v_{a})=\sigma(v_{0}v_{b})$ or
$\sigma(v_{0}v_{a})=\sigma(v_{0}v_{c})$. So,
$r_{\Sigma}(v_{a}|B_{3})=r_{\Sigma}(v_{b}|B_{3})$ or
$r_{\Sigma}(v_{a}|B_{3})=r_{\Sigma}(v_{c}|B_{3})$. Hence, $B_{3}$ does not
resolve $\Sigma$ and proceeding along the same lines, no set with cardinality
less than $k-2$ resolves $\Sigma$. By Theorem 2.2, $\dim(\Sigma)=k$ or $k-1$.
Combining all the cases, $\dim(W_{n}^{\sigma})=\dim(W_{n})$ or
$\dim(W_{n})-1$. In other words, $\operatorname{mdd}(W_{n}^{\sigma})=0$ or
$1$, which proves the theorem. ∎

The figures that follow illustrate the case when $n=9$. The thick lines in all
the figures represent positive edges and the dashed lines are negative edges.
The blackened vertices belong to the basis in the order of their suffixes.

$v_{0}$$v_{1}$$v_{2}(1,1,-2,2)$$v_{3}$$v_{4}(2,1,-2,2)$$v_{5}(-2,-2,1,-2)$$v_{6}$$(-2,-2,1,-1)v_{7}$$v_{8}$$(1,2,-2,1)v_{9}$
Figure 1: $\dim(W_{9}^{\sigma})=\dim(W_{9})=4$, i.e.,
$\operatorname{mdd}(W_{9}^{\sigma})=0$
$v_{0}$$v_{1}$$v_{2}(1,2,2)$$v_{3}(2,2,2)$$v_{4}(-2,-2,-2)$$v_{5}(-2,-1,-2)$$v_{6}$$(2,1,1)v_{7}$$v_{8}$$(1,2,1)v_{9}$
Figure 2: $\dim(W_{9}^{\sigma})=3\neq\dim(W_{9})=4$ i.e.,
$\operatorname{mdd}(W_{9}^{\sigma})=1$

##  4 Metric dimension of signed trees

We now turn our attention to find the metric dimension of signed trees. Though
we do not get a result for a general signed tree, Theorem 4.3 deals with
signed trees with some specific conditions. In the case of unsigned trees,
there is a formula for finding the metric dimension given in [2, 7] which is
stated in Theorem 4.1. But before delving into the details we shall recall
some important definitions.
A leaf of a tree is a vertex of degree $1$. A vertex of degree at least $3$ in
a tree $T$ is called a major vertex. A leaf $u$ of a tree $T$ is said to be a
terminal vertex of a major vertex $v$ of $T$ if $d(u,v)<d(u,w)$ for every
other vertex $w$ of $T$. The terminal degree $\operatorname{ter}(v)$ of a
major vertex $v$ is the number of terminal vertices of $v$. We denote the
totality of all terminal degrees in a tree $T$ by $\lambda(T)$. A major vertex
$v$ of $T$ is an exterior major vertex of $T$ if it has terminal degree
greater than 0. Number of exterior major vertices is denoted by
$\operatorname{ex}(T)$. A leg of a tree is a path from an exterior major
vertex to its terminal vertex. Let $t_{k}$ be an exterior major vertex of
$T^{\sigma}$. By $L_{k,j}$, we mean the set of all vertices on the $j^{th}$
leg of $t_{k}$ and by the index $i$ of a vertex $u_{i}\in L_{k,j}$, we mean
$d(t_{k},u_{i})=i$.

######  Theorem 4.1 ([2, 7]).

If $T$ is a tree, which is not a path, then
$\dim(T)=\lambda(T)-\operatorname{ex}(T)$.

To deal with a special class of signed trees for which we derive a formula for
their metric dimension, we define one more type of exterior major vertices as
follows.

######  Definition 4.2.

(Special exterior major vertex of a signed tree) An exterior major vertex
$t_{k}$ in a signed tree $T^{\sigma}$ is called a special exterior major
vertex if $d_{\Sigma}(t_{k},u_{i})\neq d_{\Sigma}(t_{k},v_{i})$ for all
$u_{i}\in L_{k,n}$ and $v_{i}\in L_{k,m}$, for $n\neq m$. The set of all
special exterior major vertices is denoted as $\eta(T^{\sigma})$.

######  Theorem 4.3.

Let $T^{\sigma}$ be a signed tree (which is not a path) such that
$\operatorname{ter}(t_{k})\neq 2$ for all $t_{k}\in\eta(T^{\sigma})$. Then
$\dim(T^{\sigma})=\dim(T)-|\eta(T^{\sigma})|$. In other words, in the case of
such trees $T^{\sigma}$, $\operatorname{mdd}(T^{\sigma})=|\eta(T^{\sigma})|$.

###### Proof.

Let $\\{t_{1},t_{2},...t_{n}\\}$ be the exterior major vertices of
$T^{\sigma}$ and let $W_{i}$ denote be set of the terminal vertices
$\\{t_{i1},t_{i2},...t_{im_{i}}\\}$ from $t_{i}$ for $1\leq i\leq n$. Denote
$W=\cup W_{i}$. By Theorem 4.1, $\dim(T)=\lambda(T)-\operatorname{ex}(T)$ and
the set formed from $W$ by removing one terminal vertex corresponding to each
$t_{i}$, is a basis for $T$, and by Theorem 2.1, it resolves $T^{\sigma}$
also. Now we modify this resolving set to a basis of $T^{\sigma}$ by removing
terminal vertices in a specific way as follows.
Choose an exterior major vertex $t_{k}$ of $T^{\sigma}$. Then the following
cases arise.
$\boldsymbol{Case(i)}$: $t_{k}\in\eta(T^{\sigma})$. In this case, there exist
two terminal vertices $t_{kn}$ and $t_{km}$ of $t_{k}$ such that
$d_{\Sigma}(u_{i},t_{k})\neq d_{\Sigma}(v_{i},t_{k})$ $\forall$ $u_{i}\in
L_{kn}$ and $\forall\ v_{i}\in L_{km}$. Therefore, we can remove $t_{kn}$ and
$t_{km}$ from $W$. To prove the minimality, if we remove one more terminal
vertex $t_{kl}$, say, of $t_{k}$ then
$d_{\Sigma}(t_{k},u_{1})=d_{\Sigma}(t_{k},v_{1})$ for $u_{1}\in L_{kl}$ and
$v_{1}\in L_{kn}$ or $L_{km}$. So we can remove two terminal vertices for a
$t_{k}\in\eta(T^{\sigma}$) and we fix them as $t_{km}$ and $t_{kn}$.
$\boldsymbol{Case(ii)}$: $t_{k}\notin\eta(T^{\sigma})$. Here, there exist two
legs $L_{kn}$ and $L_{km}$ of $t_{k}$ such that
$d_{\Sigma}(u_{i},t_{k})=d_{\Sigma}(v_{i},t_{k})$ for some $u_{i}\in L_{kn}$
and $v_{i}\in L_{km}$. Hence we can remove at most one terminal vertex if
$t_{k}\notin\eta(T^{\sigma}$) and choose it arbitrarily. Thus we can conclude
that if $W^{*}$ is the set formed from $W$ by removing two terminal vertices
as in Case (i), if $t_{k}\in\eta(T^{\sigma})$ and remove one terminal vertex
arbitrarily if $t_{k}\notin\eta(T^{\sigma})$, forms a basis of $T^{\sigma}$.
Hence,

$\dim(T^{\sigma})=|W^{*}|=\lambda(T)-\operatorname{ex}(T)-|\eta(T^{\sigma})|=\dim(T)-|\eta(T^{\sigma})|$

. ∎

We have $|\eta(K_{1,n}^{\sigma})|=1$, if $K_{1,n}^{\sigma}$ is non-
homogeneous. So Theorem 2.10 becomes a particular case of Theorem 4.3. We
provided a detailed proof there to show how one could select the resolving set
of a signed star.
Generally, the following bounds hold good for any signed tree.

######  Theorem 4.4.

Let $T^{\sigma}$ be a signed tree, then
$\dim(T)-\operatorname{ex}(T)\leq\dim(T^{\sigma})\leq\dim(T)$. In other words,
$\operatorname{mdd}(T^{\sigma})\leq\operatorname{ex}(T)$.

###### Proof.

From Theorem 2.2,

$\dim(T^{\sigma})\leq\ \dim(T)$ (4.1)

As in the proof of Theorem 4.3, we can remove at most two terminal vertices
with respect to each one of the exterior major vertices. Therefore, $\
T^{\sigma}$ has at least $\lambda(T)-2\operatorname{ex}(T)$ elements in its
basis. So,

$\lambda(T)-2\operatorname{ex}(T)\leq\dim(T^{\sigma})$ (4.2)

Therefore, combining results in (4.1) and (4.2),
$\dim(T)-\operatorname{ex}(T)\leq\dim(T^{\sigma})\leq\dim(T)$. ∎

The signed trees, one case of which is exhibited in the following figure, form
an interesting class with the metric dimension as small as $2$ where as the
unsinged underlying tree has a much greater metric dimension.

$u$$v$
Figure 3:  $\dim(T^{\sigma})=2;\ \ \dim(T)=7$ i.e.,
$\operatorname{mdd}(T^{\sigma})=5$

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