Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of
Graphs111This work was supported by National Natural Science Foundation of
China (Grant No. 11871256) and Gansu Provincial Department of Education: Youth
Doctoral fund project (Grant No. 2021QB-090)

Qian Qian LIU He Ping ZHANG222The corresponding author.

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.
R. China

E-mail $:$<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract: Let $G$ be a simple graph with $2n$ vertices and a perfect matching.
The forcing number $f(G,M)$ of a perfect matching $M$ of $G$ is the smallest
cardinality of a subset of $M$ that is contained in no other perfect matching
of $G$. Among all perfect matchings $M$ of $G$, the minimum and maximum values
of $f(G,M)$ are called the minimum and maximum forcing numbers of $G$, denoted
by $f(G)$ and $F(G)$, respectively. Then $f(G)\leq F(G)\leq n-1$. Che and Chen
(2011) proposed an open problem: how to characterize the graphs $G$ with
$f(G)=n-1$. Later they showed that for a bipartite graph $G$, $f(G)=n-1$ if
and only if $G$ is a complete bipartite graph $K_{n,n}$. In this paper, we
completely solve the problem of Che and Chen, and show that $f(G)=n-1$ if and
only if $G$ is a complete multipartite graph or a graph obtained from complete
bipartite graph $K_{n,n}$ by adding arbitrary edges in the same partite set.
For all graphs $G$ with $F(G)=n-1$, we prove that the forcing spectrum of each
such graph $G$ forms an integer interval by matching 2-switches and the
minimum forcing numbers of all such graphs $G$ form an integer interval from
$\lfloor\frac{n}{2}\rfloor$ to $n-1$.

Keywords: Perfect matching, minimum forcing number, maximum forcing number,
forcing spectrum, complete multipartite graph

AMS subject classification: 05C70, 05C75, 05C35

##  1 Introduction

We only consider finite and simple graphs. Let $G$ be a graph with vertex set
$V(G)$ and edge set $E(G)$. The _order_ of $G$ is the number of vertices in
$G$. A graph is _trivial_ if it contains only one vertex. Otherwise, it is
_non-trivial_. The _degree_ of vertex $v$ in $G$, written $d_{G}(v)$, is the
number of edges incident to $v$. An _isolated vertex_ is a vertex of degree 0.
The maximum degree and the minimum degree of $G$ are denoted by $\Delta(G)$
and $\delta(G)$, respectively. If all vertices of $G$ have degree $k$, then
$G$ is _$k$_ -_regular_. A complete graph of order $n$ is denoted by $K_{n}$.
Let $P_{n}$ be a path with $n$ vertices and $\overline{P_{n}}$ be the
complement of $P_{n}$.

For an edge subset $F$ of $G$, we write $G-F$ for the subgraph of $G$ obtained
by deleting the edges in $F$. If $F=\\{e\\}$, we simply write $G-e$ instead of
$G-\\{e\\}$. For a vertex subset $T$ of $G$, we write $G-T$ for the subgraph
of $G$ obtained by deleting all vertices in $T$ and their incident edges. If
$T=\\{v\\}$ is a singleton, we write $G-v$ rather than $G-\\{v\\}$. For a
vertex subset $T$ of $G$, we write $G[T]$ for the subgraph $G-(V(G)\setminus
T)$, induced by $T$. For a graph $H$, $G$ is _$H$ -free_ if it contains no $H$
as an induced subgraph.

_A perfect matching_ of a graph is a set of disjoint edges covering all
vertices of the graph. A graph $G$ is _factor-critical_ if $G-u$ has a perfect
matching for every vertex $u$ of $G$. A graph $G$ is _bicritical_ if $G$
contains an edge and $G-u-v$ has a perfect matching for every pair of distinct
vertices $u$ and $v$ in $G$. A 3-connected bicritical graph is called a
_brick_. For a nonnegative integer $l$, a connected graph $G$ with at least
$2l+2$ vertices is _$l$ -extendable_ if $G$ has a perfect matching and every
matching of size $l$ is contained in a perfect matching of $G$.

A perfect matching coincides with a Kekulé structure in organic chemistry or a
dimer covering in statistic physics. Klein and Randić [13] proposed the “
_innate degree of freedom_ ” of a Kekulé structure, i.e., the least number of
double bonds can determine this entire Kekulé structure, which plays an
important role in resonant theory. Afterwards, it was called the forcing
number by Harary et al. [10]. A _forcing set_ for a perfect matching $M$ of
$G$ is a subset of $M$ that is contained in no other perfect matching of $G$.
The smallest cardinality of a forcing set of $M$ is called the _forcing
number_ of $M$, denoted by $f(G,M)$.

Let $G$ be a graph with a perfect matching $M$. A cycle of $G$ is _M-
alternating_ if its edges appear alternately in $M$ and $E(G)\setminus M$. If
$C$ is an $M$-alternating cycle of $G$, then the symmetric difference $M\oplus
E(C):=(M\setminus E(C))\cup(E(C)\setminus M)$ is another perfect matching of
$G$. We use $V(S)$ to denote the set of all end vertices in an edge subset $S$
of $E(G)$. An equivalent condition for a forcing set of a perfect matching was
mentioned by Pachter and Kim as follows.

######  Lemma 1.1 ([18]).

Let $G$ be a graph with a perfect matching $M$. Then a subset $S\subseteq M$
is a forcing set of $M$ if and only if $G-V(S)$ contains no $M$-alternating
cycles.

Let $c(M)$ denote the maximum number of disjoint $M$-alternating cycles in
$G$. Then $f(G,M)\geq c(M)$ by Lemma 1.1. For plane bipartite graphs, Pachter
and Kim obtained the following minimax theorem.

######  Theorem 1.2 ([18]).

Let $G$ be a plane bipartite graph. Then $f(G,M)=c(M)$ for any perfect
matching $M$ of $G$.

The _minimum_ (resp. _maximum_) _forcing number_ of $G$ is the minimum (resp.
maximum) value of $f(G,M)$ over all perfect matchings $M$ of $G$, denoted by
$f(G)$ (resp. $F(G)$). Adams et al. [2] showed that determining a smallest
forcing set of a perfect matching is NP-complete for bipartite graphs with the
maximum degree 3. Using this result, Afshani et al. [3] proved that
determining the minimum forcing number is NP-complete for bipartite graphs
with the maximum degree 4. However, the computational complexity of the
maximum forcing number of a graph is still an open problem [3].

Xu et al. [29] showed that the maximum forcing number of a hexagonal system is
equal to its resonant number. The same result also holds for a polyomino graph
[39, 16] and for a BN-fullerene graph [22]. In general, for 2-connected plane
bipartite graphs, the resonant number can be computed in polynomial time (see
Ref. [1] due to Abeledo and Atkinson). Hence, the maximum forcing numbers of
such three classes of graphs can be computed in polynomial time.

Moreover, some minimax results have been obtained [38, 39]: for each perfect
matching $M$ of a hexagonal system $G$ with $f(G,M)=F(G)$, there exist $F(G)$
disjoint $M$-alternating hexagons in $G$; for every perfect matching $M$ of
polyomino graphs $G$ with $f(G,M)=F(G)$ or $F(G)-1$, $f(G,M)$ is equal to the
maximum number of disjoint $M$-alternating squares in $G$.

Zhang and Li [32], and Hansen and Zheng [9] independently determined the
hexagonal systems $G$ with $f(G)=1$, and Zhang and Zhang [35] gave a
generalization to plane bipartite graphs $G$ with $f(G)=1$. For 3-connected
cubic graphs $G$ with $f(G)=1$, Wu et al. [28] showed that it can be generated
from $K_{4}$ via $Y\rightarrow\Delta$-operation. For a convex hexagonal system
$H(a_{1},a_{2},a_{3})$ with a perfect matching, recently Zhang and Zhang [36]
proved that its minimum forcing number is equal to
min$\\{a_{1},a_{2},a_{3}\\}$ by monotone path systems.

For $n$-dimensional hypercube $Q_{n}$, Pachter and Kim [18] conjectured that
$f(Q_{n})=2^{n-2}$ for integer $n\geq 2$. Later Riddle [21] confirmed it for
even $n$ by the trailing vertex method. Recently, Diwan [8] proved that the
conjecture holds by linear algebra. Using well-known Van der Waerden theorem,
Alon showed that $F(Q_{n})>c2^{n-1}$ for any constant $0<c<1$ and sufficient
large $n$ (see [21]). There are also some researches about the minimum or
maximum forcing numbers of other graphs, such as grids [3, 18, 14, 12, 15],
fullerene graphs [11, 34, 22, 23], toroidal polyhexes [26], toroidal and Klein
bottle lattices [12, 21, 14], etc.

We denote by $\mathcal{G}_{2n}$ the set of all graphs of order $2n$ and with a
perfect matching. Let $G\in\mathcal{G}_{2n}$. Then each perfect matching of
$G$ has $n$ edges and any $n-1$ edges among it form a forcing set. So we have
that $f(G)\leq F(G)\leq n-1$. Che and Chen [6] proposed how to characterize
the graphs $G$ with $f(G)=n-1$. Afterwards, they [5] solved the problem for
bipartite graphs and obtained the following result.

######  Theorem 1.3 ([5]).

Let $G$ be a bipartite graph with $2n$ vertices. Then $f(G)=n-1$ if and only
if $G$ is complete bipartite graph $K_{n,n}$.

In this paper, we solve the problem of Che and Chen. Let
$\mathcal{K}_{n,n}^{+}$ be a family of graphs obtained by adding arbitrary
additional edges in the same partite set to the complete bipartite graph
$K_{n,n}$. In Section 2, we discuss some basic properties for graphs
$G\in\mathcal{G}_{2n}$ with $F(G)=n-1$, and obtain that $G$ is $n$-connected,
1-extendable except for graphs in $\mathcal{K}_{n,n}^{+}$. In particular, we
give a characterization for a perfect matching $M$ of $G$ with $f(G,M)=n-1$.
In Section 3, we answer the problem proposed by Che and Chen, and obtain that
$f(G)=n-1$ if and only if $G$ is either a complete multipartite graph or a
graph in $\mathcal{K}_{n,n}^{+}$. In Section 4, for all such 1-extendable
graphs $G$ with $F(G)=n-1$, we determine which of them are not 2-extendable.
Finally in Section 5 we show that $f(G)\geq\lfloor\frac{n}{2}\rfloor$ for any
graph $G\in\mathcal{G}_{2n}$ with $F(G)=n-1$, and the minimum forcing numbers
of all such graphs form an integer interval $[\lfloor\frac{n}{2}\rfloor,n-1]$.
Also we prove that the forcing spectrum (set of forcing numbers of all perfect
matchings) of each such graph $G$ forms an integer interval.

##  2 Some basic properties of graphs $G\in\mathcal{G}_{2n}$ with $F(G)=n-1$

Let $G\in\mathcal{G}_{2n}$ with $F(G)=n-1$. In this section, we will obtain
some basic properties of graph $G$. By definition of forcing numbers, we
obtain the following observation.

######  Observation 2.1.

Let $G\in\mathcal{G}_{2n}$. Then

(i) $F(G)=n-1$ if and only if $f(G,M)=n-1$ for some perfect matching $M$ of
$G$, and

(ii) $f(G)=n-1$ if and only if $f(G,M)=n-1$ for every perfect matching $M$ of
$G$.

We call a graph $G\in\mathcal{G}_{2n}$ with $F(G)=n-1$ _minimal_ if
$F(G-e)\leq n-2$ for each edge $e$ of $G$. Next we give a characterization for
a perfect matching $M$ of $G$ with $f(G,M)=n-1$.

######  Lemma 2.2.

Let $G\in\mathcal{G}_{2n}$ for $n\geq 2$. Then $G$ has a perfect matching $M$
such that $f(G,M)=n-1$ if and only if $G[V(\\{e_{i},e_{j}\\})]$ contains an
$M$-alternating cycle for any two distinct edges $e_{i}$ and $e_{j}$ of $M$.
Moreover, $G$ is minimal if and only if $G[V(\\{e_{i},e_{j}\\})]$ is exactly
an $M$-alternating 4-cycle for any two distinct edges $e_{i}$ and $e_{j}$ of
$M$.

###### Proof.

(1) Suppose that $f(G,M)=n-1$. Then $M\setminus\\{{e_{i},e_{j}}\\}$ is not a
forcing set of $M$ for any two distinct edges $e_{i}$ and $e_{j}$ of $M$. By
Lemma 1.1, $G[V(\\{e_{i},e_{j}\\})]$ contains an $M$-alternating cycle.
Conversely, for any subset $S$ of $M$ with size less than $n-1$, there are two
distinct edges in $M\setminus S$. By the assumption, $G-V(S)$ contains an
$M$-alternating cycle. By Lemma 1.1, $S$ is not a forcing set of $M$, that is,
$f(G,M)\geq n-1$.

(2) Suppose that $G$ is minimal. Since $f(G,M)=n-1$, $G[V(\\{e_{i},e_{j}\\})]$
contains an $M$-alternating 4-cycle for any two distinct edges $e_{i}$ and
$e_{j}$ of $M$. Moreover, $G[V(\\{e_{i},e_{j}\\})]$ is exactly an
$M$-alternating 4-cycle. If not, then $G[V(\\{e_{i},e_{j}\\})]$ is isomorphic
to $K_{4}$ or a graph obtained from $K_{4}$ by deleting an edge, say
$u_{i}u_{j}$, where $e_{l}=u_{l}v_{l}$ for $l=i,j$. Then
$G[V(\\{e_{i},e_{j}\\})]-{v_{i}v_{j}}$ contains an $M$-alternating cycle. So
$F(G-v_{i}v_{j})=f(G-v_{i}v_{j},M)=n-1$, which contradicts the minimality of
$G$.

Conversely, by the assumption, we have $f(G,M)=n-1$ where
$M=\\{e_{l}=u_{l}v_{l}|l=1,2,\dots,n\\}$ is a perfect matching of $G$. By
Observation 2.1, $F(G)=n-1$. Next we are to prove that $G$ is minimal. Let
$G^{\prime}=G-e$ where $e$ is an arbitrary edge of $G$. We can show that
$G^{\prime}$ has a perfect matching. If $e\notin M$, then $M$ is a perfect
matching of $G^{\prime}$. If $e=e_{i}\in M$ for some $1\leq i\leq n$, then
$G[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ is exactly an $M$-alternating 4-cycle $C$
for any $j\neq i$ and $1\leq j\leq n$. Then $M\oplus E(C)$ is a perfect
matching of $G^{\prime}$. For any perfect matching $M^{\prime}$ of
$G^{\prime}$, we will show that $f(G^{\prime},M^{\prime})\leq n-2$, and so $G$
is minimal.

Without loss of generality, any edge $e$ of $G$ can be represented as either
$u_{i}v_{i}$ or $u_{i}v_{j}$ where $j\neq i$ because any edge $u_{j}v_{j}$ can
be written as $v_{j}u_{j}$ by switching two end vertices. Let $w\in\\{u,v\\}$.

Case 1. $e=u_{i}v_{i}$.

Then $\\{u_{i}w_{j},v_{i}w_{k}\\}\subseteq M^{\prime}$ for some integers $j,k$
different from $i$. It follows that $v_{i}w_{j}\notin E(G^{\prime})\subset
E(G)$ since $G[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ is a 4-cycle by the assumption.
Then $G[\\{u_{i},w_{j},v_{i},w_{k}\\}]$ does not contain a 4-cycle of
$G^{\prime}$ since $v_{i}$ cannot be adjacent to either $u_{i}$ or $w_{j}$ in
$G^{\prime}$. By Lemma 1.1, $M^{\prime}\setminus\\{u_{i}w_{j},v_{i}w_{k}\\}$
is a forcing set of $M^{\prime}$. So, $f(G^{\prime},M^{\prime})\leq n-2$.

Case 2. $e=u_{i}v_{j}$ where $j\neq i$.

Subcase 2.1. Both $u_{i}v_{i}$ and $u_{j}v_{j}$ are contained in $M^{\prime}$.
By the assumption, $G[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ is an $M$-alternating
4-cycle of $G$ where $\\{u_{i}v_{i},u_{j}v_{j}\\}\subseteq M$. So
$G^{\prime}[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ is just a path of length three
since $G^{\prime}=G-e=G-u_{i}v_{j}$. By Lemma 1.1,
$M^{\prime}\setminus\\{u_{i}v_{i},u_{j}v_{j}\\}$ is a forcing set of
$M^{\prime}$. So, $f(G^{\prime},M^{\prime})\leq n-2$.

Subcase 2.2. At least one of $u_{i}v_{i}$ and $u_{j}v_{j}$ is not contained in
$M^{\prime}$. Without loss of generality, assume that $u_{j}v_{j}\notin
M^{\prime}.$ Since $e=u_{i}v_{j}$ is not an edge of $G^{\prime}$, there exists
a vertex $w_{l}\neq u_{i}$ such that $v_{j}w_{l}\in M^{\prime}$. Note that
$v_{i}v_{j}$ cannot be an edge of $G^{\prime}\subset G$ since
$G[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ is an $M$-alternating 4-cycle of $G$. Then
$w_{l}\neq v_{i}$. If $v_{i}w_{l}\notin E(G^{\prime})$, then $v_{i}$ is
adjacent to neither $w_{l}$ nor $v_{j}$ in $G^{\prime}$. So
$G^{\prime}[\\{v_{i},w_{t},w_{l},v_{j}\\}]$ contains no
$M^{\prime}$-alternating cycles where $v_{i}w_{t}\in M^{\prime}$. By Lemma
1.1, $M^{\prime}\setminus\\{v_{i}w_{t},v_{j}w_{l}\\}$ is a forcing set of
$M^{\prime}$. Thus, $f(G^{\prime},M^{\prime})\leq n-2$. If $v_{i}w_{l}\in
E(G^{\prime})$, then $u_{i}w_{l}\notin E(G^{\prime})\subset E(G)$ since
$G[\\{v_{i},u_{i},u_{l},v_{l}\\}]$ is exactly an $M$-alternating 4-cycle by
the assumption. Since $G^{\prime}=G-e$, we have $e=u_{i}v_{j}\notin
E(G^{\prime})$. So $u_{i}$ is adjacent to neither $w_{l}$ nor $v_{j}$ in
$G^{\prime}$, and $G^{\prime}[\\{u_{i},w_{k},w_{l},v_{j}\\}]$ contains no
$M^{\prime}$-alternating cycles where $u_{i}w_{k}\in M^{\prime}$. By Lemma
1.1, $M^{\prime}\setminus\\{u_{i}w_{k},v_{j}w_{l}\\}$ is a forcing set of
$M^{\prime}$. So, $f(G^{\prime},M^{\prime})\leq n-2$. ∎

By Lemma 2.2, adding some extra edges to a minimal graph $G$ maintains the
same maximum forcing number as $G$. Clearly, $K_{n,n}$ is minimal for $n\geq
1$ by Lemma 2.2. For $n\geq 3$, we have other such minimal graphs shown in
Fig. 1.

Fig. 1: Some examples of minimal graphs with $n=3$ and 4.

Let $G$ be a graph with a perfect matching. An edge $e$ of $G$ is called a
_fixed double bond_ if $e$ is contained in all perfect matchings of $G$. The
connectivity and edge connectivity of $G$ are denoted by $\kappa(G)$ and
$\lambda(G)$, respectively.

######  Lemma 2.3.

Assume that $G\in\mathcal{G}_{2n}$ has $F(G)=n-1>0$. Then

(i) $G$ has no fixed double bond, that is, $G-e$ has a perfect matching for
each edge $e$ of $G$.

(ii) $\kappa(G)\geq n$. Moreover, if $G$ is minimal, then $G$ is $n$-regular,
and $\kappa(G)=\lambda(G)=n$.

###### Proof.

(i) It has been implied in the sufficiency part of (2) in the proof of Lemma
2.2.

(ii) Since $F(G)=n-1$, there exists a perfect matching $M$ of $G$ such that
$f(G,M)=n-1$ where $M=\\{u_{i}v_{i}|i=1,2,\dots,n\\}$. For any $X\subseteq
V(G)$ with $|X|<n$, there is at least one pair of vertices $u_{i}$ and $v_{i}$
not in $X$ for some $1\leq i\leq n$. Then for the other vertices not in $X$,
say $u_{j}$ (resp. $v_{j}$), either $u_{i}u_{j}$ or $v_{i}u_{j}$ (resp.
$u_{i}v_{j}$ or $v_{i}v_{j}$) is contained in $E(G)$ by Lemma 2.2. Hence $G-X$
is connected and so $G$ is $n$-connected. Therefore, $\kappa(G)\geq n$.

If $G$ is minimal, then for any $1\leq i\leq n$, $u_{i}$ (resp. $v_{i}$) is
adjacent to exactly one of $u_{j}$ and $v_{j}$ for any $1\leq j\leq n$ and
$j\neq i$ by Lemma 2.2. Combining that $u_{i}v_{i}$ is an edge, we obtain that
$u_{i}$ and $v_{i}$ are of degrees $n$. So $G$ is $n$-regular. Combining that
$n\leq\kappa(G)\leq\lambda(G)\leq\delta(G)=n$, we obtain that
$\kappa(G)=\lambda(G)=n$. ∎

The number of odd components of $G$ is denoted by $o(G)$. The following result
gives an equivalent condition for a bicritical graph.

######  Lemma 2.4 ([17]).

A graph $G$ is bicritical if and only if for any $X\subseteq V(G)$ and
$|X|\geq 2$, $o(G-X)\leq|X|-2$.

Next we will show that $\mathcal{K}^{+}_{n,n}$ is a subclass of graphs
$G\in\mathcal{G}_{2n}$ with $F(G)=n-1$ such that $G$ contains an independent
set of size $n$.

######  Lemma 2.5.

Assume that $G\in\mathcal{G}_{2n}$ has $F(G)=n-1$. Then $G$ is a graph in
$\mathcal{K}^{+}_{n,n}$ if and only if $G$ contains an independent set of size
$n$. Otherwise, $G$ is a brick, and thus 1-extendable.

###### Proof.

(1) The necessity is obvious. We prove sufficiency next. Since $F(G)=n-1$,
there exists a perfect matching $M$ of $G$ such that $f(G,M)=n-1$. Let
$\\{u_{1},u_{2},\dots,u_{n}\\}$ be an independent set of size $n$ in $G$ and
let $M=\\{u_{i}v_{i}|i=1,2,\dots,n\\}$. Then $u_{i}u_{j}\notin E(G)$ for any
$1\leq i<j\leq n$. By Lemma 2.2, $\\{u_{i}v_{j},v_{i}u_{j}\\}$ is contained in
$E(G)$. Hence, $G$ is some graph in $\mathcal{K}_{n,n}^{+}$, for there may be
some other edges with both end vertices in $\\{v_{1},v_{2},\dots,v_{n}\\}$.

(2) If $G$ is not a graph in $\mathcal{K}^{+}_{n,n}$, then we will prove that
$G$ is a brick. For $n\geq 3$, $G$ is 3-connected by Lemma 2.3. For $n\leq 2$,
exactly one graph $K_{4}$ that is not in $\mathcal{K}^{+}_{n,n}$ is
3-connected. Next we prove that $G$ is bicritical. Suppose that $X\subseteq
V(G)$ with $|X|\geq 2$. If $|X|\leq n-1,$ then $G-X$ is connected by Lemma
2.3. Hence $o(G-X)\leq 1\leq|X|-1$. Otherwise, we have $|X|\geq n$. Then
$o(G-X)\leq|V(G-X)|\leq n$. Since $G$ is not a graph in
$\mathcal{K}_{n,n}^{+}$, $G$ contains no independent set of size $n$. Hence
$o(G-X)\leq n-1\leq|X|-1$. Since $G$ is of even order, $o(G-X)$ and $|X|$ are
of the same parity. So $o(G-X)\leq|X|-2$. By Lemma 2.4, $G$ is bicritical. ∎

##  3 Graphs $G\in\mathcal{G}_{2n}$ with the minimum forcing number $n-1$

In this section, we will determine all graphs $G\in\mathcal{G}_{2n}$ with
$f(G)=n-1$ to completely solve the problem proposed by Che and Chen [6].

Tutte’s theorem states that $G$ has a perfect matching if and only if
$o(G-S)\leq|S|$ for any $S\subseteq V(G)$. By Tutte’s theorem, Yu [30]
obtained an equivalent condition for a connected graph with a perfect matching
that is not $l$-extendable. Recently, for an $(l-1)$-extendable graph $G$ with
$l\geq 1$, Alajbegović et al. [4] obtained that $G$ is not $l$-extendable if
and only if there exists a subset $S\subseteq V(G)$ such that $G[S]$ contains
$l$ independent edges and $o(G-S)=|S|-2l+2$. In fact, $S$ can be chosen so
that each component of $G-S$ is factor-critical (see Theorem 2.2.3 in [7]).
Combining these, we obtain the following result.

######  Lemma 3.1.

Let $l\geq 1$ be an integer and $G$ be an $(l-1)$-extendable graph of order at
least $2l+2$. Then $G$ is not $l$-extendable if and only if there exists a
subset $S\subseteq V(G)$ such that $G[S]$ contains $l$ independent edges, all
components of $G-S$ are factor-critical, and $o(G-S)=|S|-2l+2$.

A _complete multipartite graph_ is a graph whose vertices can be partitioned
into sets so that $u$ and $v$ are adjacent if and only if $u$ and $v$ belong
to different sets of the partition. We write $K_{n_{1},n_{2},\dots,n_{k}}$ for
the complete $k$-partite graph with partite sets of sizes
$n_{1},n_{2},\dots,n_{k}$. In fact, a complete multipartite graph is a
$\overline{P_{3}}$-free graph (see Exercise 5.2.2 in [27]).

######  Lemma 3.2 ([27]).

A graph is $\overline{P_{3}}$-free if and only if it is a complete
multipartite graph.

######  Theorem 3.3.

Let $G\in\mathcal{G}_{2n}$. Then $f(G)=n-1$ if and only if $G$ is a complete
multipartite graph with each partite set having size no more than $n$ or $G$
is some graph in $\mathcal{K}_{n,n}^{+}$.

###### Proof.

Sufficiency. Suppose that $G$ is some graph in $\mathcal{K}^{+}_{n,n}$. Then
any perfect matching $M$ of $G$ is also a perfect matching of $K_{n,n}$. By
Lemma 2.2, $f(G,M)=n-1$. By the arbitrariness of $M$, $f(G)=n-1$.

Suppose that $G=K_{n_{1},n_{2},\dots,n_{k}}$ is a complete multipartite graph
where $1\leq n_{i}\leq n$ for $1\leq i\leq k$ and $k\geq 2$. First we will
show that $G$ has a perfect matching. For any nonempty subset $S$ of $V(G)$,
if $G-S$ has at least two vertices from different partite sets of $G$, then
$G-S$ is connected and $o(G-S)\leq 1\leq|S|$. Otherwise, all vertices of $G-S$
belong to one partite set of $G$. Then $|V(G-S)|\leq n$ and $|S|\geq n$. Hence
$o(G-S)\leq n\leq|S|$. For $S=\emptyset$, $o(G-S)=o(G)=|S|$. By Tutte’s
theorem, $G$ has a perfect matching.

Clearly, the result holds for $n=1$. Next let $n\geq 2$. Suppose to the
contrary that $f(G)\leq n-2$. Then there exists a perfect matching $M$ of $G$
and a minimum forcing set $S$ of $M$ such that $|S|=f(G,M)=f(G)$. By Lemma
1.1, $G-V(S)$ contains no $M$-alternating cycles. So there are two distinct
edges $\\{u_{i}v_{i},u_{j}v_{j}\\}\subseteq M\setminus S$ and
$G[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ contains no $M$-alternating cycles. That is
to say, neither $\\{u_{i}u_{j},v_{i}v_{j}\\}$ nor
$\\{u_{i}v_{j},v_{i}u_{j}\\}$ is contained in $E(G)$. Without loss of
generality, we assume that none of $u_{i}u_{j}$ and $u_{i}v_{j}$ belong to
$E(G)$. Then $G[\\{u_{i},u_{j},v_{j}\\}]$ is isomorphic to $\overline{P_{3}}$,
which contradicts Lemma 3.2.

Necessity. Suppose that $f(G)=n-1$. If $G$ is a complete multipartite graph,
then each partite set of $G$ has size no more than $n$ for $G$ has a perfect
matching. If $G$ is not a complete multipartite graph, then by Lemma 3.2, $G$
contains an induced subgraph $H$ isomorphic to $\overline{P_{3}}$. Note that
the edge $e$ of $H$ is not in any perfect matching of $G$. Otherwise, there is
a perfect matching $M$ of $G$ containing $e$. By Observation 2.1,
$f(G,M)=n-1$. Let $v$ be the vertex of $H$ except for both end vertices of
$e$, and $e^{\prime}$ be the edge of $M$ incident with $v$. Then
$G[V(\\{e,e^{\prime}\\})]$ contains no $M$-alternating cycles for $v$ is not
incident with any end vertices of $e$, which contradicts Lemma 2.2. So $G$ is
not 1-extendable. By Lemma 3.1, there exists $S\subseteq V(G)$ such that
$G[S]$ contains an edge, all components of $G-S$ are factor-critical, and
$o(G-S)=|S|\geq 2$.

We claim that all components of $G-S$ are singletons. Otherwise, assume that
$C_{1}$ is a non-trivial component of $G-S$. Let $M$ be a perfect matching of
$G$. By Observation 2.1, $f(G,M)=n-1$. Since $o(G-S)=|S|$, $M$ matches $S$ to
distinct components of $G-S$. Assume that $e_{1}$ is an edge in $M\cap
E(C_{1})$ and $e_{2}$ is an edge of $M$ which connects a vertex of $S$ and a
vertex of another component $C_{2}$ of $G-S$. Then $G[V(\\{e_{1},e_{2}\\})]$
contains no $M$-alternating cycles, which contradicts Lemma 2.2. Therefore,
each component of $G-S$ is a singleton and so $|S|+o(G-S)=2n$. It follows that
$o(G-S)=n$ since we have shown that $o(G-S)=|S|$. So $G$ contains an
independent set of size $n$. By Lemma 2.5, $G$ is a graph in
$\mathcal{K}^{+}_{n,n}$. ∎

Taking $n=3$ for example,
$K_{3,3},K_{3,2,1},K_{3,1,1,1},K_{2,2,2},K_{2,2,1,1},K_{2,1,1,1,1},K_{6}$ are
all complete multipartite graphs with each partite set having size no more
than 3 and $K_{3,3}+e$ is the unique graph in $\mathcal{K}^{+}_{3,3}$ except
for above complete multipartite graphs, where $e$ is an edge connecting any
two nonadjacent vertices of $K_{3,3}$.

##  4 Extendability of graphs $G\in\mathcal{G}_{2n}$ with $F(G)=n-1$

Let $G$ be a graph in $\mathcal{G}_{2n}$ with $F(G)=n-1$ and be different from
graphs in $\mathcal{K}_{n,n}^{+}$. By Lemma 2.5, $G$ is 1-extendable. However,
it is not necessarily 2-extendable. We know that an $l$-extendable graph is
$(l-1)$-extendable for an integer $l\geq 1$, and 2-extendable graphs are
either bricks or braces (2-extendable bipartite graphs) [19]. In this section,
we will determine which graphs in Lemma 2.5 are 1-extendable but not
2-extendable.

######  Theorem 4.1.

Let $G\in\mathcal{G}_{2n}$ with $F(G)=n-1$ where $n\geq 3$. Then $G$ is
1-extendable but not 2-extendable if and only if $G$ has a perfect matching
$M=\\{u_{i}v_{i}|i=1,2,\dots,n\\}$ with $f(G,M)=n-1$ so that one of the
following conditions holds.

(i) $n\geq 4$, $G[\\{v_{1},v_{2},\dots,v_{n}\\}]$ consists of one triangle and
$n-3$ isolated vertices, and $G[\\{u_{1},u_{2},\dots,u_{n}\\}]$ has at least
two independent edges (see an example in Fig. 2(a)).

(ii) $\\{v_{1},v_{2},\dots,v_{n-1}\\}$ is an independent set,
$G[\\{u_{1},u_{2},\dots,u_{n},v_{n}\\}]$ has at least two independent edges,
and $\\{v_{i}v_{n},v_{j}u_{n}\\}\subseteq E(G)$ for some $i$ and $j$ with
$1\leq i,j\leq n-1$ (see examples (b) and (c) in Fig. 2).

Fig. 2: Three examples for non-2-extendable graphs.

###### Proof.

Sufficiency. Since $f(G,M)=n-1$, by Lemma 2.5, $G$ is 1-extendable or $G$
contains an independent set of size $n$. We claim that $G$ contains no
independent set of size $n$. If we have done, then $G$ is 1-extendable. Let
$S=\\{u_{1},u_{2},\dots,u_{n}\\}$ (resp.
$\\{u_{1},u_{2},\dots,u_{n},v_{n}\\}$) be a subset of $V(G)$ corresponding to
(i) (resp. (ii)). Then $G[S]$ contains two independent edges, all components
of $G-S$ are factor-critical, and $o(G-S)=|S|-2$. By Lemma 3.1, $G$ is not
2-extendable.

Now we prove the claim. Let $I$ be any independent set of $G$. We will prove
that $|I|\leq n-1$. We consider the graphs $G$ satisfying (i). Let
$\\{v_{1},v_{2},\dots,v_{n-3}\\}$ be the set of $n-3$ isolated vertices and
$G[\\{v_{n-2},v_{n-1},v_{n}\\}]$ be the triangle of
$G[\\{v_{1},v_{2},\dots,v_{n}\\}]$. If
$I\subseteq\\{u_{1},u_{2},\dots,u_{n}\\}$, then $|I|\leq n-2$ for
$G[\\{u_{1},u_{2},\dots,u_{n}\\}]$ has at least two independent edges.
Otherwise, there exists $v_{i}\in I$ for some $1\leq i\leq n$. For any $1\leq
j\leq n$ and $j\neq i$, $G[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ contains an
$M$-alternating 4-cycle by Lemma 2.2. So if $v_{i}v_{j}\notin E(G)$ for some
$j$ then $v_{i}u_{j}\in E(G)$ and $u_{j}\notin I$. Hence, if $1\leq i\leq n-3$
then $u_{j}\notin I$ for any $1\leq j\leq n$ and $j\neq i$ since
$v_{i}v_{j}\notin E(G)$. Hence $I\subseteq\\{v_{1},v_{2},\dots,v_{n}\\}$ and
$|I|\leq n-2$. If $n-2\leq i\leq n$, then $u_{j}\notin I$ for any $1\leq j\leq
n-3$ since $v_{i}v_{j}\notin E(G)$. Hence
$I\subseteq\\{v_{1},v_{2},\dots,v_{n-3},v_{i}\\}\cup(\\{u_{n-2},u_{n-1},u_{n}\\}\setminus\\{u_{i}\\})$.
If $v_{1}\in I$, then $\\{u_{n-2},u_{n-1},u_{n}\\}\cap I=\emptyset$ by Lemma
2.2, so $|I|\leq n-2$. Otherwise, $v_{1}\notin I$. Then $|I|\leq n-1$.

We consider the graphs $G$ satisfying (ii). If
$I\subseteq\\{u_{1},u_{2},\dots,u_{n},v_{n}\\}$, then $|I|\leq n-1$ for
$G[\\{u_{1},u_{2},\dots,u_{n},v_{n}\\}]$ has at least two independent edges.
Otherwise, there exists $v_{k}\in I$ for some $1\leq k\leq n-1$. Since
$v_{k}v_{l}\notin E(G)$ for any $1\leq l\leq n-1$ and $l\neq k$,
$v_{k}u_{l}\in E(G)$ since $G[\\{u_{k},v_{k},u_{l},v_{l}\\}]$ contains an
$M$-alternating 4-cycle by Lemma 2.2. So $u_{l}\notin I$ and
$I\subseteq\\{v_{1},v_{2},\dots,v_{n},u_{n}\\}$. By the assumption,
$\\{v_{j}u_{n},v_{i}v_{n}\\}\subseteq E(G)$. If $i\neq j$, then
$\\{v_{j}u_{n},v_{i}v_{n}\\}$ are two independent edges and $|I|\leq n-1$.
Otherwise, $G[\\{v_{i},v_{n},u_{n}\\}]$ is a triangle and $|I|\leq n-1$.

Necessity. By Lemma 3.1, there exists a subset $S\subseteq V(G)$ such that
$G[S]$ contains two independent edges, all components of $G-S$ are factor-
critical, and $o(G-S)=|S|-2$. Hence $|S|\geq 4$. Let $C_{i}~{}(1\leq
i\leq|S|-2)$ be all components of $G-S$. Since $F(G)=n-1$, there exists a
perfect matching $M$ of $G$ so that $f(G,M)=n-1$. We claim that $G-S$ has at
most one component containing exactly three vertices, and the others are
trivial. Otherwise, $G-S$ has one component, say $C_{1}$, containing at least
five vertices or $G-S$ has two components, say $C_{1}$ and $C_{2}$, containing
exactly three vertices. Since $G-S$ has exactly $|S|-2$ factor-critical
components, in either case there is one edge $e_{1}$ in $M$ belonging to
$C_{1}$ or $C_{2}$. Then none of end vertices of $e_{1}$ are adjacent to any
vertex, say $w$, of another component. So $G[V(\\{e_{1},e_{2}\\})]$ cannot
contain an $M$-alternating cycle, where $e_{2}$ is the edge in $M$ incident
with $w$, which contradicting Lemma 2.2.

If one component of $G-S$ is a triangle and the other components are
singletons, then $|S|=n\geq 4$. Let $V(C_{i})=\\{v_{i}\\}$ for each $1\leq
i\leq n-3$, $V(C_{n-2})=\\{v_{n-2},v_{n-1},v_{n}\\}$ and $u_{i}v_{i}\in M$ for
each $1\leq i\leq n-3$. Since $f(G,M)=n-1$, each edge of $C_{n-2}$ does not
belong to $M$ by Lemma 2.2. So we denote the remaining three edges of $M$ by
$u_{n-2}v_{n-2},u_{n-1}v_{n-1}$ and $u_{n}v_{n}$. Thus
$M=\\{u_{i}v_{i}|i=1,2,\dots,n\\}$ and $S=\\{u_{1},u_{2},\dots,u_{n}\\}$.
Hence (i) holds.

If all components of $G-S$ are trivial, then $|S|=n+1$ and $o(G-S)=n-1$. Let
$V(C_{i})=\\{v_{i}\\}$ and $u_{i}v_{i}\in M$ for each $1\leq i\leq n-1$. Then
the remaining edge of $M$ is denoted by $u_{n}v_{n}$. So
$\\{v_{1},v_{2},\dots,v_{n-1}\\}$ is an independent set of $G$ and
$S=\\{u_{1},u_{2},\dots,u_{n},v_{n}\\}$ with $G[S]$ containing at least two
independent edges. Since $G$ is not a graph in $\mathcal{K}_{n,n}^{+}$, $G$
contains no independent set of size $n$ by Lemma 2.5. Combining that
$\\{v_{1},v_{2},\dots,v_{n-1}\\}$ is an independent set of $G$, there exists
$i$ and $j$ with $1\leq i,j\leq n-1$ such that $\\{u_{n}v_{j},v_{n}v_{i}\\}$
is contained in $E(G)$. So (ii) holds. ∎

Next we will determine all minimal non-2-extendable graphs in Theorem 4.1.

######  Corollary 4.2.

Let graph $G$ in Theorem 4.1 be minimal. Then $G$ is not 2-extendable if and
only if (ii) in Theorem 4.1 holds where $i\neq j$,
$\\{u_{1},u_{2},\dots,u_{n-1}\\}$ is an independent set and
$G[\\{u_{n},v_{n},u_{i},v_{i}\\}]$ is just a 4-cycle for $1\leq i\leq n-1$
(see an example in Fig. 2(c)).

###### Proof.

Sufficiency. It suffices to prove that $G$ is minimal by Theorem 4.1. Since
$f(G,M)=n-1$, by Lemma 2.2, $G[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ contains an
$M$-alternating 4-cycle for any $1\leq i<j\leq n-1$. By the assumption,
$\\{u_{1},u_{2},\dots,u_{n-1}\\}$ and $\\{v_{1},v_{2},\dots,v_{n-1}\\}$ are
two independent sets. So neither $v_{i}v_{j}$ nor $u_{i}u_{j}$ is an edge of
$G$. Hence $G[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ is exactly a 4-cycle. Combining
that $G[\\{u_{n},v_{n},u_{i},v_{i}\\}]$ is just a 4-cycle for $1\leq i\leq
n-1$, we obtain that $G$ is minimal by Lemma 2.2.

Necessity. By Theorem 4.1, $G$ has a perfect matching
$M=\\{u_{i}v_{i}|i=1,2,\dots,n\\}$ with $f(G,M)=n-1$ so that (i) or (ii)
holds. First we show that each graph $G$ satisfying (i) is not minimal. Let
$\\{v_{1},v_{2},\dots,v_{n-3}\\}$ be the set of $n-3$ isolated vertices and
$G[\\{v_{n-2},v_{n-1},v_{n}\\}]$ be the triangle of
$G[\\{v_{1},v_{2},\dots,v_{n}\\}]$. Then there is at least one vertex, say
$u_{i}$ for some $1\leq i\leq n-3$, incident with one of the two independent
edges of $G[\\{u_{1},u_{2},\dots,u_{n}\\}]$. Assume that $u_{i}u_{j}$ is such
an edge for some $1\leq j\leq n$. By Lemma 2.2,
$G[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ contains an $M$-alternating 4-cycle. Since
$v_{i}v_{j}\notin E(G)$, we have $\\{u_{i}v_{j},v_{i}u_{j}\\}\subseteq E(G)$.
But $u_{i}u_{j}\in E(G)$, $G[\\{u_{i},v_{i},u_{j},v_{j}\\}]$ is not a 4-cycle.
So $G$ is not minimal by Lemma 2.2.

Next let $G$ be a graph satisfying (ii). Since $G$ is minimal,
$G[\\{u_{k},v_{k},u_{l},v_{l}\\}]$ is just a 4-cycle for $1\leq k<l\leq n$ by
Lemma 2.2. Since $\\{v_{i}v_{n},v_{j}u_{n}\\}\subseteq E(G)$ for some $i$ and
$j$ with $1\leq i,j\leq n-1$, we have $i\neq j$. Since
$\\{v_{1},v_{2},\dots,v_{n-1}\\}$ is an independent set, $v_{k}v_{l}\notin
E(G)$ for $1\leq k<l\leq n-1$. By Lemma 2.2,
$G[\\{u_{k},v_{k},u_{l},v_{l}\\}]$ is a 4-cycle $u_{k}v_{l}u_{l}v_{k}u_{k}$
and $u_{k}u_{l}\notin E(G)$. Hence $\\{u_{1},u_{2},\dots,u_{n-1}\\}$ is an
independent set. ∎

##  5 Minimum forcing numbers and forcing spectrum of graphs $G$ with
$F(G)=n-1$

The _forcing spectrum_ of a graph $G$ is the set of forcing numbers of all
perfect matchings of graph $G$. If the forcing spectrum is an integer
interval, then we say it is _continuous_ (or _consecutive_). Afshani et al.
[3] showed that any finite subset of positive integers is the forcing spectrum
of some graph. Besides, they [3] obtained that the forcing spectra of column
continuous subgrids are continuous by matching 2-switches. Further, Zhang and
Jiang [33] generalized their result to any polyomino with perfect matchings by
applying the $Z$-transformation graph. Zhang and Deng [31] obtained that the
forcing spectrum of any hexagonal system with a forcing edge form either the
integer interval from 1 to its Clar number or with only the gap 2. For more
researches on the forcing spectra of special graphs, see [2, 3, 11, 25, 20,
24, 37].

Let $G\in\mathcal{G}_{2n}$ with $F(G)=n-1$. In this section we will prove that
$f(G)\geq\lfloor\frac{n}{2}\rfloor$, and find that for the class of such
graphs all minimum forcing numbers of them form an integer interval
$[\lfloor\frac{n}{2}\rfloor,n-1]$. Further, we will show that the forcing
spectrum of each such graph $G$ is continuous. Next we give a lemma obtained
by Che and Chen.

######  Lemma 5.1 ([5]).

Let $G$ be a $k$-connected graph with a perfect matching. Then
$f(G)\geq\lfloor\frac{k}{2}\rfloor$.

Combining Lemmas 2.3 and 5.1, we have the following result.

######  Corollary 5.2.

Let $G\in\mathcal{G}_{2n}$ with $F(G)=n-1$. Then
$f(G)\geq\lfloor\frac{n}{2}\rfloor$.

For $0\leq k\leq\lfloor\frac{n-1}{2}\rfloor$, let $H_{k}\in\mathcal{G}_{2n}$
be a minimal graph with $f(H_{k},M_{0})=n-1$, where
$M_{0}=\\{u_{i}v_{i}|i=1,2,\dots,n\\}$ is a perfect matching of $H_{k}$, so
that $H_{k}[\\{u_{1},u_{2},\dots,u_{n}\\}]$ contains exactly $k$ edges
$\\{u_{2i-1}u_{2i}|i=1,2,\dots,k\\}$. Then
$H_{k}[\\{v_{1},v_{2},\dots,v_{n}\\}]$ contains exactly $k$ edges
$\\{v_{2i-1}v_{2i}|i=1,2,\dots,k\\}$ and
$H_{k}[\\{u_{2k+1},v_{2k+1},\dots,u_{n},v_{n}\\}]$ is isomorphic to a complete
bipartite graph (see examples in Fig. 3).

Fig. 3: $H_{k}$ with $k=\lfloor\frac{n-1}{2}\rfloor$ and $n=6$ and 7.

######  Remark 5.3.

For $0\leq k\leq\lfloor\frac{n-1}{2}\rfloor$, $f(H_{k})\leq n-k-1.$
Especially, for $k=\lfloor\frac{n-1}{2}\rfloor$ we have
$f(H_{k})=\lfloor\frac{n}{2}\rfloor$, i.e., the lower bound of Corollary 5.2
is sharp.

Let
$M=\\{u_{2i-1}u_{2i},v_{2i-1}v_{2i}|i=1,2,\dots,k\\}\cup\\{u_{2k+1}v_{2k+1},u_{2k+2}v_{2k+2},\dots,u_{n}v_{n}\\}$
be a perfect matching of $H_{k}$. It follows that
$S=\\{v_{2i-1}v_{2i}|i=1,2,\dots,k\\}\cup\\{u_{2k+2}v_{2k+2},u_{2k+3}v_{2k+3},\\\
\dots,u_{n}v_{n}\\}$ is a forcing set of $M$ by Lemma 1.1 (see examples in
Fig. 3, where bold lines form $S$). So $f(H_{k})\leq
f(H_{k},M)\leq|S|=k+(n-2k-1)=n-k-1$.

Especially, for $k=\lfloor\frac{n-1}{2}\rfloor$ we have $f(H_{k})\leq
n-k-1=\lfloor\frac{n}{2}\rfloor$. Combining Corollary 5.2, we obtain that
$f(H_{k})=\lfloor\frac{n}{2}\rfloor$.

Next we will prove that $f(H_{k})=n-k-1$ for $0\leq
k\leq\lfloor\frac{n-1}{2}\rfloor$. Here we give two simple facts that can be
obtained from Lemma 1.1.

######  Fact 5.4.

If $G^{\prime}$ is a spanning subgraph of $G$ with $f(G^{\prime},M)=|S|$ for
some perfect matching $M$ of $G^{\prime}$, then $f(G,M)\geq|S|$.

######  Fact 5.5.

Let $M$ be a perfect matching of $G$ with $M=M_{1}\cup M_{2}$, and let
$G_{i}=G[V(M_{i})]$ for $i=1,2$. Then $f(G,M)\geq
f(G_{1},M_{1})+f(G_{2},M_{2})$.

######  Lemma 5.6.

For $0\leq k\leq\lfloor\frac{n-1}{2}\rfloor$, $f(H_{k})=n-k-1$.

###### Proof.

By Remark 5.3, it suffices to prove that $f(H_{k})\geq n-k-1$. We proceed by
induction on $n$. It is trivial for $n=1$. Since $H_{0}\cong K_{n,n}$, we have
$f(H_{0})=n-1$ by Theorem 1.3. By Remark 5.3, for
$k=\lfloor\frac{n-1}{2}\rfloor$ we have
$f(H_{k})=\lfloor\frac{n}{2}\rfloor=n-k-1$. So next we suppose $n\geq 2$ and
$1\leq k\leq\frac{n-3}{2}$. Let $M$ be any perfect matching of $H_{k}$. Then
we have the following claims.

Claim 1. If $u_{i}v_{i}\in M$ for some $1\leq i\leq 2k$, then $f(H_{k},M)\geq
n-k-1$.

Let $H^{2}=H_{k}-\\{u_{i},v_{i}\\}$ and $M_{2}=M\setminus\\{u_{i}v_{i}\\}$.
Then $H^{2}$ is isomorphic to $H^{\prime}_{k^{\prime}}$ with $n^{\prime}=n-1$
and $k^{\prime}=k-1$. By the induction hypothesis,

$f(H^{2},M_{2})=f(H^{\prime}_{k^{\prime}},M_{2})\geq
f(H^{\prime}_{k^{\prime}})\geq n^{\prime}-k^{\prime}-1=n-k-1.$

By Fact 5.5, $f(H_{k},M)\geq f(H^{2},M_{2})\geq n-k-1$.

Claim 2. If $G$ has an $M$-alternating 4-cycle containing exactly two edges of
$\\{u_{2i-1}u_{2i},\\\ v_{2i-1}v_{2i}|i=1,2,\dots,k\\}$, then $f(H_{k},M)\geq
n-k-1$.

If $M\cap\\{u_{2i-1}u_{2i},v_{2i-1}v_{2i}|i=1,2,\dots,k\\}=\emptyset$, then we
assume that $C$ is an $M$-alternating 4-cycle of $G$, and
$\\{u_{2i-1}u_{2i},v_{2j-1}v_{2j}\\}$ is contained in $C$ for some $1\leq
i,j\leq k$. So $C=u_{2i-1}v_{2j-1}v_{2j}u_{2i}u_{2i-1}$ or
$u_{2i-1}v_{2j}v_{2j-1}u_{2i}u_{2i-1}$. Let $H^{2}=H_{k}-V(C)$ and
$M_{2}=M\cap E(H^{2})$, $M_{1}=M\setminus M_{2}$, $H^{1}=H_{k}[V(M_{1})]$.
Then $H^{1}$ contains $C$ and $f(H^{1},M_{1})\geq 1$ by Theorem 1.2. On the
other hand, $H^{2}-\\{v_{2i-1}v_{2i},u_{2j-1}u_{2j}\\}$ is isomorphic to
$H^{\prime}_{k^{\prime}}$ with $n^{\prime}=n-2$, $k^{\prime}\leq k-1$, and
$M_{2}$ is a perfect matching of $H^{\prime}_{k^{\prime}}$. By the induction
hypothesis and Fact 5.4,

$f(H^{2},M_{2})\geq f(H^{\prime}_{k^{\prime}},M_{2})\geq f(H_{k^{\prime}})\geq
n^{\prime}-k^{\prime}-1\geq n-k-2.$

By Fact 5.5, $f(H_{k},M)\geq f(H^{1},M_{1})+f(H^{2},M_{2})\geq
1+(n-k-2)=n-k-1.$

If $M\cap\\{u_{2i-1}u_{2i},v_{2i-1}v_{2i}|i=1,2,\dots,k\\}\neq\emptyset$, then
the intersection of the two sets is denoted by $M_{1}$. By the structure of
$H_{k}$, all vertices of $\\{u_{1},u_{2},\dots,u_{n}\\}\setminus V(M_{1})$
must match into all vertices of $\\{v_{1},v_{2},\dots,v_{n}\\}\setminus
V(M_{1})$. So we have

$|M_{1}\cap\\{u_{2j-1}u_{2j}|j=1,2,\dots,k\\}|=|M_{1}\cap\\{v_{2j-1}v_{2j}|j=1,2,\dots,k\\}|.$

Let $M_{2}=M\setminus M_{1}$ and $H^{i}=H_{k}[V(M_{i})]$ for $i=1$, 2. For any
pair of edges $\\{u_{2i-1}u_{2i},v_{2j-1}v_{2j}\\}$ of $M_{1}$ where $1\leq
i,j\leq k$, $G[\\{u_{2i-1},u_{2i},v_{2j-1},v_{2j}\\}]$ is either a 4-cycle or
$K_{4}$ by the structure of $H_{k}$. So
$G[\\{u_{2i-1},u_{2i},v_{2j-1},v_{2j}\\}]$ contains an $M_{1}$-alternating
4-cycle, and so $H^{1}$ contains $\frac{|M_{1}|}{2}$ disjoint
$M_{1}$-alternating 4-cycles. By Lemma 1.1,
$f(H^{1},M_{1})\geq\frac{|M_{1}|}{2}$. On the other hand,

$H^{2}-\\{u_{2j-1}u_{2j}|v_{2j-1}v_{2j}\in
M_{1},j=1,2,\dots,k\\}\cup\\{v_{2j-1}v_{2j}|u_{2j-1}u_{2j}\in
M_{1},j=1,2,\dots,k\\}$

is isomorphic to $H^{\prime}_{k^{\prime}}$ with
$n^{\prime}=n-|M_{1}|,~{}k^{\prime}\leq k-\frac{|M_{1}|}{2}$. Since

$k^{\prime}\leq
k-\frac{|M_{1}|}{2}\leq\frac{n-3-|M_{1}|}{2}<\frac{n-2-|M_{1}|}{2}\leq\lfloor\frac{n-1-|M_{1}|}{2}\rfloor=\lfloor\frac{n^{\prime}-1}{2}\rfloor,$

by the induction hypothesis and Fact 5.4, we obtain that

$f(H^{2},M_{2})\geq f(H^{\prime}_{k^{\prime}},M_{2})\geq
f(H^{\prime}_{k^{\prime}})\geq n^{\prime}-k^{\prime}-1\geq
n-k-1-\frac{|M_{1}|}{2}.$

By Fact 5.5, we obtain that

$f(H_{k},M)\geq
f(H^{1},M_{1})+f(H^{2},M_{2})\geq\frac{|M_{1}|}{2}+(n-k-1-\frac{|M_{1}|}{2})=n-k-1.$

By Claims 1 and 2, from now on we suppose that $M$ contains no $u_{i}v_{i}$
for any $1\leq i\leq 2k$ and any $M$-alternating 4-cycle of $G$ contains at
most one edge of $\\{u_{2i-1}u_{2i},v_{2i-1}v_{2i}|i=1,2,\dots,k\\}$.
Particularly, $M$ contains no edges of
$\\{u_{2i-1}u_{2i},v_{2i-1}v_{2i}|i=1,2,\dots,k\\}$. So we may assume
$\\{v_{1}u_{3},v_{2}u_{5}\\}$ is contained in $M$. Next we are going to
consider the edge of $M$ incident with $v_{3}$, say $v_{3}u_{l}$. Then $l$ has
four possible values: $l\leq 2$, $l=6$, $7\leq l\leq 2k$ and $l\geq 2k+1$.
Furthermore, if $7\leq l\leq 2k$, then we suppose $l=7$ and continue to
consider the edge of $M$ incident with $v_{5}$. Until we obtain an edge
$v_{h}u_{l}\in M$ so that one of the other three cases ($l\leq h-1$, $l=h+3$,
$l\geq 2k+1$) holds (since $H_{k}$ is finite, such edge exists).

Fig. 4: Illustration for the proof of Lemma 5.6, where bold lines form
$M_{1}$.

If $l=h+3$, then let $M_{1}=\\{v_{1}u_{3},v_{2}u_{5},\dots,v_{h}u_{h+3}\\}$
(see an example in Fig. 4(a) where $h=5$), $M_{2}=M\setminus M_{1}$, and
$H^{i}=H_{k}[V(M_{i})]$ for $i=1,2$. Then $H^{1}$ contains an
$M_{1}$-alternating 4-cycle $v_{1}u_{5}v_{2}u_{3}v_{1}$. By Theorem 1.2,
$f(H^{1},M_{1})\geq 1$. On the other hand,
$H^{2}-\\{u_{1}u_{2},v_{h+2}v_{h+3}\\}$ is isomorphic to
$H^{\prime}_{k^{\prime}}$ with $n^{\prime}=n-|M_{1}|$. Since contribution of
the edges $v_{1}u_{3}$, $v_{h}u_{h+3}$ to $k^{\prime}$ is -2, 0, and that of
each other edges in $M_{1}$ is -1, we have $k^{\prime}=k-|M_{1}|$. Since

$k^{\prime}=k-|M_{1}|\leq\frac{n-3-2|M_{1}|}{2}<\frac{n-2-|M_{1}|}{2}\leq\lfloor\frac{n-1-|M_{1}|}{2}\rfloor=\lfloor\frac{n^{\prime}-1}{2}\rfloor,$

by the induction hypothesis and Fact 5.4,

$f(H^{2},M_{2})\geq f(H^{\prime}_{k^{\prime}},M_{2})\geq
f(H^{\prime}_{k^{\prime}})\geq n^{\prime}-k^{\prime}-1=n-k-1.$

By Fact 5.5, $f(H_{k},M)\geq f(H^{1},M_{1})+f(H^{2},M_{2})\geq 1+n-k-1=n-k$.

If $l\in[2k+1,n]\cup[1,h-1]$ (see an example in Fig. 4(b) where $h=5,~{}l=2)$,
then we continue to consider the edge of $M$ incident with $v_{h+2}$, say
$v_{h+2}u_{i}$. Then $i$ has three possible values: $i\leq h+1$, $h+4\leq
i\leq 2k$ and $i\geq 2k+1$. Furthermore, if $h+4\leq i\leq 2k$, then we
suppose $i=h+4$ and continue to consider the edge of $M$ incident with
$v_{h+4}$. Until we obtain an edge of $M$, say $v_{r}u_{t}$ with
$t\in[2k+1,n]\cup[1,r-1]$ (By the finiteness of $G$, such edge exists) (see an
example in Fig. 4(b) where $r=9,~{}t=11)$. Let
$M_{1}=\\{v_{1}u_{3},v_{2}u_{5},\dots,v_{h}u_{l},\dots,v_{r}u_{t}\\}$,
$M_{2}=M\setminus M_{1}$, and $H^{i}=H_{k}[V(M_{i})]$ for $i=1,2$. Then
$H^{2}-u_{1}u_{2}$ is isomorphic to $H^{\prime}_{k^{\prime}}$ with
$n^{\prime}=n-|M_{1}|$. Since contribution of the edges $v_{1}u_{3}$,
$v_{h}u_{l}$, $v_{r}u_{t}$ to $k^{\prime}$ is -2, 0, 0 and that of each other
edges in $M_{1}$ is -1, we have $k^{\prime}=k-(|M_{1}|-1)$. Since

$k^{\prime}=k-|M_{1}|+1\leq\frac{n-1-2|M_{1}|}{2}\leq\frac{n-2-|M_{1}|}{2}\leq\lfloor\frac{n-1-|M_{1}|}{2}\rfloor=\lfloor\frac{n^{\prime}-1}{2}\rfloor,$

by the induction hypothesis and Fact 5.4,

$f(H^{2},M_{2})\geq f(H^{\prime}_{k^{\prime}},M_{2})\geq
f(H^{\prime}_{k^{\prime}})\geq n^{\prime}-k^{\prime}-1=n-k-2.$

By Fact 5.5, $f(H_{k},M)\geq f(H^{1},M_{1})+f(H^{2},M_{2})\geq
1+(n-k-2)=n-k-1$.

By the arbitrariness of $M$, we obtain that $f(H_{k})\geq n-k-1$. ∎

Combining Lemma 5.6 and Corollary 5.2, we obtain the following result.

######  Theorem 5.7.

All minimum forcing numbers of graphs $G\in\mathcal{G}_{2n}$ with $F(G)=n-1$
form an integer interval $[\lfloor\frac{n}{2}\rfloor,n-1]$.

Suppose that $M$ is a perfect matching of $G$. If $C$ is an $M$-alternating
cycle of length 4, then $M\oplus E(C)$ is a _matching 2-switch_ on $M$.
Afshani et al. [3] obtained that a matching 2-switch on a perfect matching
does not change the forcing number by more than 1.

######  Lemma 5.8 ([3]).

If $M$ is a perfect matching of $G$ and $C$ is an $M$-alternating cycle of
length 4, then

$|f(G,M\oplus E(C))-f(G,M)|\leq 1.$

By Lemma 5.8, if $M_{1},M_{2},\dots,M_{s}$ is a sequence of perfect matchings
such that $M_{i+1}$ is obtained from $M_{i}$ by a matching 2-switch for $1\leq
i\leq s-1$, then the integer interval [min$\\{f(G,M_{1}),f(G,M_{s})\\}$,
max$\\{f(G,M_{1}),f(G,M_{s})\\}]$ is contained in the forcing spectrum of $G$.

######  Theorem 5.9.

If $G\in\mathcal{G}_{2n}$ with $F(G)=n-1$, then its forcing spectrum is
continuous.

###### Proof.

Let $M_{s}=\\{u_{i}v_{i}|i=1,2,\dots,n\\}$ be a perfect matching of $G$ with
$f(G,M_{s})=n-1$. Then we will prove that $M_{s}$ can be obtained from any
perfect matching $M$ of $G$ by repeatedly applying matching 2-switches. If we
have done, then $M_{s}$ can be obtained from a perfect matching of $G$ with
the minimum forcing number by repeatedly applying matching 2-switches. So the
forcing spectrum of $G$ is continuous by Lemma 5.8.

We proceed by induction on $n$. For $n=1$, $G\cong K_{2}$ and the result is
trivial. Next, for $n\geq 2$, $f(G,M_{s})=n-1>0$. Take any perfect matching
$M$ of $G$ different from $M_{s}$. Then we have the following claims.

Claim 1. If $M_{s}\cap M\neq\emptyset$, then $M_{s}$ can be obtained from $M$
by repeatedly applying matching 2-switches.

Suppose $M_{s}\cap M$ contains an edge $u_{i}v_{i}$ for some $1\leq i\leq n$.
Let $G^{\prime}=G-\\{u_{i},v_{i}\\}$ and
$M^{\prime}=M\setminus\\{u_{i}v_{i}\\}$,
$M_{s}^{\prime}=M_{s}\setminus\\{u_{i}v_{i}\\}$. Then $G^{\prime}$ has
$2(n-1)$ vertices, $M^{\prime}$ and $M_{s}^{\prime}$ are two distinct perfect
matchings of $G^{\prime}$. By Lemma 2.2, $f(G^{\prime},M_{s}^{\prime})=n-2$.
By the induction hypothesis, $M_{s}^{\prime}$ can be obtained from
$M^{\prime}$ by repeatedly applying matching 2-switches. Hence, $M_{s}$ is
also obtained from $M$ by the same series of matching 2-switches, and the
claim holds.

Claim 2. If $M_{s}\oplus M$ contains a cycle of length 4, then $M_{s}$ can be
obtained from $M$ by repeatedly applying matching 2-switches.

Assume that $C$ is a cycle of length 4 contained in $M_{s}\oplus M$ and
$V(C)=\\{u_{i},v_{i},u_{j},v_{j}\\}$. Let $M^{\prime}=M\oplus E(C)$. Then
$\\{u_{i}v_{i},u_{j}v_{j}\\}\subseteq M_{s}\cap M^{\prime}$. By Claim 1,
$M_{s}$ can be obtained from $M^{\prime}$ by repeatedly applying matching
2-switches. Hence $M_{s}$ can be obtained from $M$ by repeatedly applying
matching 2-switches, and the claim holds.

By Claims 1 and 2, from now on we suppose that $M_{s}\cap M=\emptyset$ and
$M_{s}\oplus M$ contains no cycles of length 4. Then we may suppose that
$v_{1}u_{2}\in M$. So $u_{1}v_{2}\notin M$. Without loss of generality, we
assume that $u_{1}v_{3}\in M$. If $u_{2}v_{3}$ is an edge of $G$, then
$C_{1}=u_{1}v_{1}u_{2}v_{3}u_{1}$ is an $M$-alternating 4-cycle. So
$M^{\prime}=M\oplus E(C_{1})$ is a perfect matching of $G$ and $M_{s}\cap
M^{\prime}=\\{u_{1}v_{1}\\}$. From Claim 1, we are done. Otherwise,
$u_{2}v_{3}$ is not an edge of $G$. By Lemma 2.2,
$\\{u_{2}u_{3},v_{2}v_{3}\\}\subseteq E(G)$ (see Fig. 5(a)). Next we consider
the following two cases according as whether $v_{2}u_{3}$ belongs to $M$ or
not.

Case 1. $v_{2}u_{3}\in M$. If $v_{1}u_{3}\in E(G)$, then
$C_{2}=v_{1}u_{2}v_{2}u_{3}v_{1}$ is an $M$-alternating 4-cycle. So
$M^{\prime}=M\oplus E(C_{2})$ is a perfect matching of $G$ and $M_{s}\cap
M^{\prime}=\\{u_{2}v_{2}\\}$. From Claim 1, we are done. Otherwise,
$v_{1}u_{3}\notin E(G)$. By Lemma 2.2, $\\{u_{1}u_{3},v_{1}v_{3}\\}\subseteq
E(G)$ (see Fig. 5(b)). Then $C_{3}=u_{1}u_{3}v_{2}v_{3}u_{1}$ is an
$M$-alternating 4-cycle and $C_{4}=v_{2}v_{3}v_{1}u_{2}v_{2}$ is an $M\oplus
E(C_{3})$-alternating 4-cycle. So $M^{\prime}=M\oplus E(C_{3})\oplus E(C_{4})$
is a perfect matching of $G$ that is obtained from $M$ by two matching
2-switches and $M_{s}\cap M^{\prime}=\\{u_{2}v_{2}\\}$. From Claim 1, we are
done.

Case 2. $v_{2}u_{3}\notin M$. Without loss of generality, we can suppose that
$v_{2}u_{4}\in M$. If $v_{1}u_{4}$ is an edge of $G$, then
$C_{5}=v_{1}u_{2}v_{2}u_{4}v_{1}$ is an $M$-alternating 4-cycle. So
$M^{\prime}=M\oplus E(C_{5})$ is a perfect matching of $G$ and $M_{s}\cap
M^{\prime}=\\{u_{2}v_{2}\\}$. From Claim 1, we are done. Otherwise,
$v_{1}u_{4}$ is not an edge of $G$. By Lemma 2.2,
$\\{u_{1}u_{4},v_{1}v_{4}\\}\subseteq E(G)$ (see Fig. 5(c)). Next we
distinguish the following two subcases according to $u_{3}v_{4}\in M$ or not.

Fig. 5: Illustration for the proof of Theorem 5.9.

Subcase 2.1. $u_{3}v_{4}\in M$ (see Fig. 5(d)). Then
$C_{6}=u_{1}v_{3}v_{2}u_{4}u_{1}$ and $C_{7}=v_{1}v_{4}u_{3}u_{2}v_{1}$ are
two $M$-alternating 4-cycles. By two matching 2-switches we have that
$M^{\prime}=M\oplus E(C_{6})\oplus E(C_{7})$ is a perfect matching of $G$.
Hence,
$M^{\prime}=(M\setminus\\{u_{1}v_{3},v_{2}u_{4},v_{1}u_{2},u_{3}v_{4}\\})\cup\\{v_{2}v_{3},u_{1}u_{4},u_{2}u_{3},v_{1}v_{4}\\}$.
So $M_{s}\oplus M^{\prime}$ contains a 4-cycle $u_{2}u_{3}v_{3}v_{2}u_{2}$.
From Claim 2, $M_{s}$ can be obtained from $M^{\prime}$ by repeatedly applying
matching 2-switches, so we are done.

Subcase 2.2. $u_{3}v_{4}\notin M$. Then we may suppose $u_{3}v_{5}\in M$. If
$u_{1}v_{5}$ is an edge of $G$, then $C_{8}=u_{1}v_{5}u_{3}v_{3}u_{1}$ is an
$M$-alternating 4-cycle. So $M^{\prime}=M\oplus E(C_{8})$ is a perfect
matching of $G$ and $M_{s}\cap M^{\prime}=\\{u_{3}v_{3}\\}$. From Claim 1, we
are done. So we may suppose $u_{1}v_{5}\notin E(G)$. By Lemma 2.2,
$\\{v_{1}v_{5},u_{1}u_{5}\\}\subseteq E(G)$ (see Fig. 5(e)). Then
$C_{9}=u_{2}u_{3}v_{5}v_{1}u_{2}$ and $C_{6}$ are two $M$-alternating
4-cycles. By two matching 2-switches we have that $M^{\prime}=M\oplus
E(C_{9})\oplus E(C_{6})$ is a perfect matching of $G$. Hence,
$M^{\prime}=(M\setminus\\{u_{1}v_{3},v_{2}u_{4},v_{1}u_{2},u_{3}v_{5}\\})\cup\\{v_{2}v_{3},u_{1}u_{4},u_{2}u_{3},v_{1}v_{5}\\}$.
So $M_{s}\oplus M^{\prime}$ contains a 4-cycle $u_{2}u_{3}v_{3}v_{2}u_{2}$.
From Claim 2, we are done. ∎

Acknowledgments

The authors are grateful to anonymous reviewers for giving valuable comments
and suggestions in improving the manuscript.

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Probing the CP nature of the Higgs coupling in $t\overline{t}h$ events at the
LHC

E. Gouveia1***Funded by Fundação para a Ciência e Tecnologia, through grant
PD/BD/128231/2016
also on behalf of
S.P. Amor dos Santos2, M.C.N. Fiolhais2,4, R. Frederix5, R. Gonçalo3, R.
Martins1, A. Onofre1, C.M.Pease4, H. Peixoto 6, A. Reigoto1, R. Santos7,8, J.
Silva6
1 LIP, Departamento de Física, Universidade do Minho, 4710-057 Braga, Portugal
2 LIP, Departamento de Física, Universidade de Coimbra, 3004-516 Coimbra,
Portugal
3 LIP, Av. Prof. Gama Pinto, n.2, 1649-003 Lisboa, Portugal;
Faculdade de Ciências da Universidade de Lisboa, Campo Grande, 1749-016
Lisboa, Portugal
4 Science Department, Borough of Manhattan Community College, City University
of New York, 199 Chambers St, New York, NY 10007, USA
5 Technische Universität München, James-Franck-Str. 1, D-85748 Garching,
Germany
6 Centro de Física, Universidade do Minho, Campus de Gualtar, 4710-057 Braga,
Portugal
7 Instituto Superior de Engenharia de Lisboa - ISEL, 1959-007 Lisboa, Portugal
8 Centro de Física Teórica e Computacional, Faculdade de Ciências,
Universidade de Lisboa, Campo Grande, Edifício C8 1749-016 Lisboa, Portugal

> The determination of the $CP$ nature of the top quark Yukawa coupling is
> addressed in this phenomenological work, using $t\overline{t}h$ events at
> the $\sqrt{s}=13$ TeV LHC, with $h\rightarrow b\overline{b}$ and through the
> dilepton decay channel of $t\overline{t}$. Signal events were generated for
> pure $CP$-even, pure $CP$-odd and mixed-$CP$ coupling scenarios. Standard
> Model (SM) backgrounds were also considered. Angular distributions and
> asymmetries were studied and found to be potentially good probes to the $CP$
> nature of the coupling, even after detector simulation and full event
> reconstruction using a kinematic fit. Expected limits to the
> $t\overline{t}h$ production cross-section times branching ratio were
> obtained for a range of top quark Yukawa $CP$-mixing angles and for
> different integrated luminosities of the LHC.

> PRESENTED AT
>
>
>
>
> TOP2017 - 10th International Workshop on Top Quark Physics
> Braga, Portugal, September 17–22, 2017

##  1 Introduction

The $CP$ nature of the couplings to gauge bosons of the recently discovered
particle compatible with the 125 GeV Higgs boson [1, 2] has been studied by
ATLAS and CMS. So far, no equivalent analyses have been carried out in the
fermionic sector alone. The top quark, due to its expected large Yukawa
coupling, will be the obvious candidate for such measurements. The top quark
Yukawa interaction with the Higgs boson can be parametrised as

$\mathcal{L}=\kappa y_{t}\overline{t}(\cos\alpha+i\gamma_{5}\sin\alpha)th,$
(1)

where the SM is recovered for $\alpha=0$ and $\kappa=1$, and
$\alpha\notin\\{0,\pi\\}$ implies a non-zero $CP$-odd component in the
coupling. The Higgs associated production process $t\overline{t}h$ allows a
direct measurement of this coupling. While a non-zero $CP$-odd component is
expected to affect the inclusive $t\overline{t}h$ production cross-section, it
has been suggested [3, 4, 5, 6] that differential cross-section measurements,
exploiting the distinct kinematics and spin correlations among intermediate
and final-state particles, can enhance sensitivity to its presence. We show
that angular distributions of the kind introduced in [7] are sensitive to the
$CP$ nature of the coupling. Also, a survey of previously proposed observables
resulted in a reduced set of those which still show significant discrimination
power even after detector simulation, event selection and event reconstruction
via a kinematic fit [8].

##  2 Event generation, simulation and reconstruction

Events from $pp$ collisions were generated at $\sqrt{s}=13$ TeV using
MadGraph5_aMC@NLO [9]. Signal $t\overline{t}h$ events, as well as the
irreducible background $t\overline{t}b\overline{b}$, were generated at NLO in
QCD. Signal samples in which the Higgs has a non-zero $CP$-odd component were
generated using the HC_NLO_X0 UFO model [6], with $\cos\alpha$ varying from -1
to 1. Particle decays were handled by MadSpin [3]. The SM backgrounds
$t\overline{t}+$jets, $t\overline{t}+V(V=Z,W)$, single top, $V+$jets,
$V+b\overline{b}$ and $VV$ were also generated. Parton shower and
hadronisation were performed with Pythia6 [10] and followed by a fast detector
simulation with the Delphes [11] package.

Events were selected with at least four jets (of which at least three must be
$b$-tagged) and exactly two opposite-charge leptons, all with $p_{T}\geq 20$
GeV and $|\eta|<2.5$. Additionally, the dilepton mass $m_{\ell\ell}$ must meet
the requirement $|m_{\ell\ell}-91\textrm{ GeV}|>10\textrm{ GeV}$. Kinematic
reconstruction of the event starts with the choice of the most likely
assignment of jets to $b$-quarks from $t$, $\overline{t}$ and $h$ decay, using
a boosted decision tree (BDT). Then, a solution for the neutrino momenta is
obtained by combining information from missing $E_{T}$ with a constraint on
$W$ and $t$ masses.

##  3 Angular distributions

Let $\Delta\theta^{t\overline{t}h}(t,h)$ (respectively
$\Delta\theta^{t\overline{t}h}(\overline{t},h)$) be the angle between the
direction of $h$ and the direction of $t$ (respectively $\overline{t}$), both
evaluated in the $t\overline{t}h$ rest-frame. Figure 1 shows two-dimensional
distributions of signal events for these angles, calculated at NLO after
parton-shower effects, prior to detector simulation and event selection.

|  |
---|---|---
Figure 1: Parton-level distributions of
$\Delta\theta^{t\overline{t}h}(\overline{t},h)$ ($x$-axis) and
$\Delta\theta^{t\overline{t}h}(t,h)$ ($y$-axis). The SM signal ($h=H$)
distribution (left) and the pure $CP$-odd scenario signal ($h=A$) distribution
(right) are shown.

In the SM scenario (left) the Higgs boson tends to travel preferentially along
a direction close to that of one of the top quarks and nearly opposite to the
other. In the pure $CP$-odd scenario (right), events are more evenly spread
over the allowed range.

The angular observables introduced in [7] are constructed from angles
$\theta^{X}_{Y}$, defined as the angle between the direction of $Y$, in the
rest frame of $X$, and the direction of $X$ in the rest frame of its parent
system. In Figure 2, normalised distributions of
$\sin(\theta^{t\overline{t}h}_{t})\sin(\theta^{h}_{W^{+}})$ and
$\sin(\theta^{t\overline{t}h}_{h})\sin(\theta^{t\overline{t}}_{\overline{t}})$
are shown after cuts and kinematic reconstruction for the SM signal, pure
$CP$-odd scenario signal and irreducible background
$t\overline{t}b\overline{b}$ events. These are among the most sensitive
observables of this kind with respect to the $CP$ nature of the signal.

From the survey of observables proposed in previous theoretical or
phenomenological studies, the ones found to be more sensitive to the different
$CP$ scenarios, after cuts and reconstruction, were
$\beta_{b\overline{b}}\Delta\theta^{lh}(\ell^{+},\ell^{-})$ , proposed in [4],
and $b_{4}=p_{t}^{z}p_{\overline{t}}^{z}/p_{t}p_{\overline{t}}$, proposed in
[5]. Distributions of these observables are shown, after cuts and
reconstruction, in Figure 2, for signal events in the SM and in the pure
$CP$-odd scenarios and for the $t\overline{t}b\overline{b}$ background.

|  |
---|---|---
|  |
Figure 2: Normalised distributions of signal events in the SM
($t\overline{t}H$) and pure $CP$-odd ($t\overline{t}A$) scenarios and of the
$t\overline{t}b\overline{b}$ background, after cuts and reconstruction.

Overall, $b_{4}$ is found to be the most sensitive observable, with associated
“forward-backward” asymmetries of +0.16 for SM signal, -0.17 for pure $CP$-odd
signal and 0.12 for $t\overline{t}b\overline{b}$.

##  4 Expected limits on $t\overline{t}h$ production cross-section

A signal-background classification BDT, combining the most sensitive
observables, was trained for each value of $\cos\alpha$. Its output was used
as a discriminant for extracting the expected 95% CL upper limits on the
production cross-section times the $h\rightarrow b\overline{b}$ branching
ratio ($\sigma\times\textrm{BR}$), under the background-only hypothesis.
Figure 3 shows, on the left, these limits as a function of $\cos\alpha$, for 3
different integrated luminosities and, on the right, the corresponding limits
on signal strength, computed relatively to the prediction for that value of
$\alpha$.

|  |
---|---|---
Figure 3: Expected 95% CL limits in the background-only scenario as a function
of $\alpha$ and for 3 integrated luminosities. Left:
$\sigma\times$BR($h\rightarrow b\overline{b}$); right:
$\mu=\sigma/\sigma_{\alpha}$.

From the left plot, we see that the analysis becomes more sensitive as the
$CP$-odd component of the coupling increases. Thus, a search optimized for the
SM $t\overline{t}h$ search is not insensitive to production through a
mixed-$CP$ coupling. With respect to CP measurements, it was recently
suggested that the single-lepton channel is nearly twice as sensitive as the
dilepton channel [12] and it would seem beneficial to combine the two.

ACKNOWLEDGEMENTS

I am thankful to the International Advisory Committee for granting me the
opportunity to present and discuss this work at the Workshop, and to the Local
Organizing Committee for a very nicely organized gathering.

## References

  * [1] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 716 (2012) 1
  * [2] S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 716 (2012) 30
  * [3] P. Artoisenet et al., JHEP 1303 (2013) 015
  * [4] F. Boudjema et al., Phys. Rev. D 92 (2015) no.1, 015019
  * [5] J. F. Gunion and X. G. He, Phys. Rev. Lett. 76 (1996) 4468
  * [6] F. Demartin et al., Eur. Phys. J. C 74 (2014) no.9, 3065
  * [7] S. P. Amor dos Santos et al., Phys. Rev. D 92 (2015) no.3, 034021
  * [8] S. Amor Dos Santos et al., Phys. Rev. D 96 (2017) no.1, 013004
  * [9] J. Alwall et al., JHEP 1407 (2014) 079
  * [10] T. Sjostrand, S. Mrenna and P. Z. Skands, JHEP 0605 (2006) 026
  * [11] J. de Favereau et al. [DELPHES 3 Collaboration], JHEP 1402 (2014) 057
  * [12] D. Azevedo, A. Onofre, F. Filthaut and R. Gonçalo, arXiv:1711.05292 [hep-ph].


# Investigation of the Gamma-Ray Bursts prompt emission under the
relativistically expanding fireball scenario

Soumya Gupta∗ Homi Bhabha National Institute, Mumbai, Maharashtra, India
Bhabha Atomic Research Center, Mumbai, Maharashtra, India  Sunder
Sahayanathan+ Homi Bhabha National Institute, Mumbai, Maharashtra, India
Bhabha Atomic Research Center, Mumbai, Maharashtra, India

###### Abstract

The spectral properties of a composite thermal emission arising from a
relativistic expanding fireball can be remarkably different from the Planck
function. We perform a detailed study of such a system to explore the features
of the prompt emission spectra from the gamma-ray bursts (GRBs). Particularly,
we address the effect of optical opacity and its dependence on the density
profile between the expanding gas and the observer. This results in a
nontrivial shape of the photospheric radius which in combination with the
constraints derived from the equal-arrival-time can result in a mild broader
spectrum compared to the Planck function. Further, we show the time-integrated
spectrum from the expanding fireball deviates significantly from the
instantaneous emission and is capable of explaining the observed broad
spectral width of the GRBs. We also show, that the demand of the spectral
width of the order of unity, obtained through statistical analysis, is
consistent with the scenario where the dynamics of the expanding fireball are
governed predominantly by the energy content of the matter.

Gamma-ray bursts–X-ray transient sources–Radiative processes

##  1 Introduction

Gamma-Ray Bursts (GRB) are the brightest events that spate the radiation in a
brief duration of time. They carry tremendous energy lasting from a few ms to
a hundred seconds, which outshines any sources in the sky. This short-lived
burst of radiation, bright in $\gamma$-ray and hard X-ray, is referred to as
the prompt phase of GRB. Even after decades of study using several space-based
missions, the radiation mechanism of the prompt phase is still under debate
(Rees & Mészáros, 2005; Pe’er, 2008; Beloborodov, 2011; Zhang et al., 2014).
Among various models, the competitive ones are the decoupled thermal emission
associated with the expansion of an initial optically thick fireball (Cavallo
& Rees, 1978; Paczynski, 1986; Goodman, 1986; Shemi & Piran, 1990; Piran et
al., 1993) and the non-thermal origin due to synchrotron loss from an electron
distribution accelerated at the shock fronts (Rees & Meszaros, 1994; Tavani,
1996; Cohen & Piran, 1997; Sari & Piran, 1997; Bošnjak et al., 2009; Daigne et
al., 2011; Zhang & Yan, 2011). Nevertheless, under the fireball scenario, both
these emission products are viable since after the release of the initial
thermal energy, the matter coasts relativistically developing shocks which
dissipate the bulk energy of the flow (Thompson, 1994; Ghisellini & Celotti,
1999; Lazzati & Begelman, 2010; Lazzati et al., 2011; Beloborodov, 2012).

The dynamics of an expanding fireball are mainly governed by the factor
$\eta$, the ratio of the energy content in radiation $E_{0}$ to the total rest
mass energy of the matter $M_{0}c^{2}$ at the beginning of the burst (Shemi &
Piran, 1990; Piran et al., 1993). In the initial phase of evolution, the
fireball is optically thick and expands freely at the expense of its internal
energy ($\eta\gg 1$), with the bulk Lorentz factor ($\Gamma$) of the flow
proportional to the radius ($\Gamma\sim R$) (Goodman, 1986). The temperature
of the fireball at this phase will largely remain constant. The acceleration
phase continues till $\eta\sim 1$ and from there on, the energy content of the
matter dominates the evolution of the fireball. During this matter-dominant
phase ($\eta\ll 1$), the fireball coasts with a constant bulk velocity and the
temperature scales as $R^{-2/3}$ (Piran et al., 1993).

In the course of evolution, the temperature of the fireball drops and once it
falls below the pair production threshold, the radiation decouples from the
matter, releasing a burst of radiation. The radius of the fireball where the
last scattering surface of the trapped photons falls is referred to as
photospheric radius $R_{ph}$.Interestingly, $R_{ph}$ can fall either during
$\eta>1$ or $\eta<1$ phase and the observed burst property will vary
significantly under these two scenarios. When $\eta>1$, the matter is still in
the accelerating phase and the instantaneous Doppler boosting nullifies the
adiabatic loss. Hence, the observed burst luminosity will be equal to that of
the initial outflow (Mészáros & Rees, 2000). On the other hand, when $\eta<1$,
the flow incurs significant adiabatic loss and the observed burst luminosity
will be less than the initial outflow.

The effect of the viewing angle and the relativistic flow have a substantial
role in the observed shape of the photospheric surface of the burst. For a
spherically symmetric expanding matter, the photospheric radius measured by a
distant observer $R_{ph}(\theta)$ will be larger for the higher viewing angle
(high latitude) of the emission cone. This will result in a non-trivial
concave shape of the photospheric surface (§2.1)(Pe’er (2008), Meng et al.
(2018)). Similarly, due to the relativistic expansion, the observed
instantaneous emission released from a certain radius has to travel a long
distance in case of higher latitude compared to the on-axis emission. Hence,
the surface constructed from the photons reaching the observer at the same
time (equal-time-surface ) will deviate from a spherical shape. If the
expansion is associated with a fall in the temperature of the fireball, then
the equal-time-surface  will be a superposition of multiple temperatures.
These effects will significantly modify the initial observed emission profile
and the spectrum of the GRB prompt phase (§2.2).

Irrespective of the radiative model, the prompt emission of GRBs is often well
represented by an empirical function involving a broken power-law with an
exponential turnover, commonly referred to as the Band function (Band et al.,
1993). This function is useful to identify the peak frequency, the low energy
and the high energy spectral indices which can be readily compared with the
constraints derived from the radiative models (Preece et al., 2002; Pe’er &
Zhang, 2006; Uhm & Zhang, 2014). Besides these, it also helps to understand
the statistical property of a large sample of GRBs, for example, the
correlation between the peak energy and the isotropic luminosity (Amati et
al., 2002; Yonetoku et al., 2004; Liang et al., 2010). Another study involving
a large sample of GRBs is their spectral width (FWHM: $\mathcal{W}$)
distribution derived from the best-fit Band function parameters. For long GRBs
the most probable $\mathcal{W}$is found to be $\sim 1.07$ estimated from a
sample of 1873 GRBs and in the case of short GRBs, it was $\sim 0.86$ obtained
from a sample of 419 GRBs (Axelsson & Borgonovo, 2015). Interestingly, these
widths are much broader than that of a Planck function (FWHM $\approx$ 0.54)
while much narrower than a typical synchrotron spectrum (FWHM $\sim$ 1.6).

In an attempt to explain the unusual $\mathcal{W}$ of GRBs, Bharali et al.
(2017) considered a multi-temperature blackbody spectrum associated with an
evolving fireball. Nevertheless, the $\mathcal{W}$ of $\sim$ 1 demanded an
unphysical faster rate of temperature evolution and this disfavoured the
fireball interpretation of GRB prompt emission. In this work, we revisit the
expanding fireball scenario of GRB prompt emission and show that a proper
inclusion of the radial evolution is capable of explaining the observed
$\mathcal{W}$ of GRBs. The effect of the concave photospheric surface and the
emission cone limited by the relativistic effects are also considered in
studying the temporal evolution of the fireball blackbody spectrum.

##  2 EXPANDING FIREBALL

###  2.1 Evolutionary Phase

The optical depth of a relativistically expanding spherical symmetric wind
measured for a photon propagating towards an observer will be (Abramowicz et
al., 1991)

$\displaystyle\tau(r,\theta)=\frac{r_{d}}{\pi r}(\theta-\beta\sin\theta)$ (1)

where, $r$ is projected distance and $\theta$ is the angle with respect to the
line of sight such that $r\,\sin\theta$ is the radial distance $R$ of the
emitted photon from the centre of the spherical wind. The quantity $\beta$ is
the velocity of expansion of the wind, assumed to be constant and expressed in
the units of $c$, and $r_{d}$ is the radius of the opaque disk measured by the
observer for photons emerging from infinity. The observed photospheric radius
will then be (Pe’er, 2008)

$\displaystyle
R_{ph}(\theta)=\frac{R_{ph0}}{(1-\beta)}\left(\frac{\theta}{\sin\theta}-\beta\right)$
(2)

where, $R_{ph0}\approx r_{d}/2\pi\Gamma^{2}$ is the minimum photospheric
radius measured at $\theta=0$ and $\Gamma$ is the bulk Lorentz factor of the
plasma. Interestingly, $R_{ph}$ will be an increasing function of $\theta$ and
hence, the observer will visualize the photospheric surface as concave though
the expansion is spherically symmetric (Figure 1).

Due to spherical expansion, the photons emitted on-axis will arrive earlier
than those emitted at higher latitudes. Hence, the photons received by an
observer at an instant will be a combination of higher latitude emissions,
when the expanding plasma is young relative to the on-axis emission. The
evolutionary equal-time-surface corresponding to the instantaneous emission by
the observer can then be obtained as (Bharali et al., 2017)

$\displaystyle R(\theta)$
$\displaystyle=R(0)\left(\frac{1-\beta}{1-\beta\cos\,\theta}\right)$ (3)

which is a decreasing function of $\theta$ and will be an ellipsoidal surface
(Figure 1). Due to relativistic expansion, the emitted photons will be
confined within a cone of a semi-vertical angle $1/\Gamma$. Hence, this angle
limits the maximum higher latitude emission received by the observer with
$R(\theta)|_{\rm max}\approx R(0)/(1+\beta)$.

Figure 1: The plot depicts the equal-time-surface in red-dashed line,
photospheric radius $R_{ph}(\theta)$ in teal-solid line and emission cone of
angle $1/\Gamma$ in black-dotted line for the $\Gamma$ of 800. The $R_{ph0}$
is scaled to unity and the grey-dot-dashed line is the axis for $\theta=0$.

During the initial stage of evolution, the opening angle can be significantly
be smaller than $1/\Gamma$ and will be governed by $R_{ph}(\theta)$. Since
$R(\theta)\geq R_{ph}(\theta)$ and from equations 2 and 3 we obtain

$\displaystyle\frac{r_{d}}{\pi
R(0)(1-\beta)}\left(\frac{\theta}{\sin\theta}-\beta\right)(1-\beta\cos\theta)\leq
1$ (4)

For a given $R(0)$, the opening angle of the emission cone can be obtained by
solving the above transcendental equation which will be smaller than
$1/\Gamma$. The on-axis radial distance beyond which the angle of the emission
cone will be constant at $1/\Gamma$ will be $R(0)\gtrsim(1+\beta)R_{ph0}$.

As the fireball expands beyond $R_{ph}(\theta)$, the photons decouple from the
inner regions and have a finite probability to be scattered along the line of
sight (Pe’er, 2008; Pe’er & Ryde, 2011; Beloborodov, 2011). The observed
intensity of the radiation will then be proportional to

$\displaystyle\zeta(\theta)$
$\displaystyle=\int\limits^{R(\theta)}_{R_{in}}\exp[{-\tau(z,\theta)}]dz$ (5)

where, the direction of $z$ is along the line of sight and $R_{in}={\rm
MIN}\\{[R(\theta)-R_{ph}(\theta)],R_{ph0}\,\cos(\theta)\\}$. Using equations 1
and 2, we obtain

$\displaystyle\zeta(\theta)=\int\limits^{R(\theta)}_{{R_{in}}}\exp\left[{\frac{-\cos(\theta)\,R_{ph}(\theta)}{z}}\right]dz$
(6)

The observed temperature of the fireball evolves depending on the energetics
of the burst phase. In the radiation dominated phase ($\eta\gg 1$), $T_{\rm
obs}\sim{\rm constant}$; while, in the matter dominated phase ($\eta\ll 1$),
$T_{\rm obs}\propto R^{-2/3}$. If we represent the on-axis temperature of the
fireball as $T_{0}$, then the temperature gradient observed over the equal-
time-surface  will be

$\displaystyle\frac{T_{\theta}}{T_{0}}$
$\displaystyle=\left[\frac{R(0)}{R(\theta)}\right]^{\alpha}$
$\displaystyle=\left(\frac{1-\beta\cos\,\theta}{1-\beta}\right)^{\alpha}$ (7)

where, $\alpha$ varies from 0 to 2/3. In terms of the on-axis photospheric
temperature $T_{ph0}$, the temperature gradient can be expressed as

$\displaystyle\frac{T_{\theta}}{T_{ph0}}=\left(\frac{R_{ph0}}{R(0)}\frac{1-\beta\cos\theta}{1-\beta}\right)^{\alpha}$
(8)

###  2.2 Observed spectrum

The instantaneous flux from the expanding fireball measured by an observer
located at a distance $D_{L}$ will be

$\displaystyle F_{\nu}$
$\displaystyle=\frac{2\pi}{D_{L}^{2}}\,\int\limits^{1}_{\cos\phi}I_{\theta}(\nu)\,R(\theta)^{2}\,\zeta(\theta)\,\mu
d\mu$ $\displaystyle=\frac{2\pi
R(0)^{2}(1-\beta)^{2}}{D_{L}^{2}}\int\limits^{1}_{\cos\phi}\frac{I_{\theta}(\nu)}{(1-\beta\mu)^{2}}\,\zeta(\theta)\,\mu
d\mu$ (9)

where, $I_{\theta}(\nu)$ is the observed specific intensity and
$\mu=\cos\theta$. Using Lorentz invariance, $I_{\theta}(\nu)$ can be expressed
in terms of the expanding gas comoving specific intensity
$I^{{}^{\prime}}_{\theta}(\nu^{{}^{\prime}})$ as 111All the quantities
represented with ′ are measured in the expanding gas comoving frame.

$\displaystyle I_{\theta}(\nu)$
$\displaystyle=I^{{}^{\prime}}_{\theta}(\nu^{{}^{\prime}})\left(\frac{\nu}{\nu^{{}^{\prime}}}\right)^{3}$
$\displaystyle=\frac{2h}{c}\frac{\nu^{3}}{\exp{\left(\frac{h\nu}{kT_{\theta}}\right)}-1}$
(10)

Figure 2: The plot represents the spectrum ($\alpha=0.6$) for the time-
averaged case ($\nu\,\mathscr{F}_{\nu}$) in a red-solid line, the
instantaneous case ($\nu\,F_{\nu}$) in an teal-dashed line and the black body
in a black-dotted line.

and using equations 6, 9 and 10, the observed instantaneous flux will be

$\displaystyle F_{\nu}$ $\displaystyle=4\pi
h\nu^{3}\left[\frac{(1-\beta)R(0)}{cD_{L}}\right]^{2}\times$
$\displaystyle\int\limits^{1}_{\cos\phi}\frac{\mu}{(1-\beta\mu)^{2}}\frac{\zeta(\theta)\,d\mu}{\left\\{\exp\left[\frac{h\nu}{kT_{ph0}}\left(\frac{R(0)}{R_{ph0}}\frac{1-\beta}{1-\beta\mu}\right)^{\alpha}\right]-1\right\\}}$
(11)

In Figure 2, we show the instantaneous spectrum from the fireball along with
an equivalent Planck function for $\alpha$ ranging from 0 to 2/3. The
temperature variation will be more pronounced for the time-averaged spectrum
and will significantly deviate from the Planck function. Let us assume the
evolution of the fireball for a duration $\Delta t$ during which the on-axis
radius expands from $R_{1}$ to $R_{2}$ ($=R_{1}+2\beta c\Gamma^{2}\Delta t$).
The time-averaged spectrum during $\Delta t$ will be equivalent to

$\displaystyle\mathscr{F}_{\nu}=\frac{\int\limits^{R_{2}}_{R_{1}}F_{\nu}(x)\,dx}{R_{2}-R_{1}}$
(12)

In Figure 2, we show the time-averaged spectrum for a duration of 5s along
with the Planck function. The spectral shape has a strong dependence on
$\alpha$ (in Figure 3) and in Figure 4, we show the variation in the peak of
the spectrum with respect to various model parameters. To study the variation
in the spectrum due to an individual parameter, the rest of the parameters
were fixed at $\Gamma=100$, $R_{ph0}=12.0$ log cm, $T_{ph0}=500$ keV, duration
of burst $\Delta t$ = 5.0 sec and $\alpha=0.67$ (Figure 3, 4 and 5). We note
the flux at the spectral peak enhances nearly by an order of ten (in Figure
3), when $\alpha$ is varied from 0 to 2/3. However, the dynamic range of the
detector will typically be of the order of 2-3. Therefore, the range of
$\alpha$ that could be studied by an instrument will be $\Delta\alpha\sim
0.15$.

Figure 3:  The plot represents the change in the time-integrated spectrum for
the $\alpha$ ranging from 0 to $2/3$, color-mapped on the right axis. The
vertical grey-dashed line represents the energy range of the NaI detector of
the Gamma-Ray Burst Monitor (GBM) onboard Fermi.

###  2.3 Spectral width

The time-averaged spectrum from the fireball can imitate various spectral
shapes with the proper choice of $\alpha$. Accordingly, their spectral width
at half the maximum flux ($\mathcal{W}$, defined as
$\mathcal{W}$=log($\nu_{2}$/$\nu_{1}$), can vary with $\alpha$. Besides this,
the variation in other parameters may also impact $\mathcal{W}$. To
investigate this, we estimated $\mathcal{W}$ for a wide range of model
parameters and the results are shown in Figure 4. It is evident the variation
in $\mathcal{W}$ is not prominent in the case of the parameters $\Gamma$,
$R_{ph0}$, $\Delta t$ and $T_{ph0}$ (The flux at half maximum is shown as a
grey-dashed line in Figure 4). However, significant change in W can be
witnessed even with moderate variation in $\alpha$.

The $\mathcal{W}$ of the order of unity, supported by the observations
(Axelsson & Borgonovo, 2015), can be obtained only for the case when
$\alpha\sim 0.65$ (which is close to 2/3) and shown as yellow triangle in the
top-left panel of Figure 5. Interestingly, besides providing an explanation
for the observed spectral width of GRBs, these results also suggest that most
of the radiation from the GRB is released during the matter-dominated phase of
the expanding fireball.

Figure 4: The plot exhibits the evolution of the time-integrated spectrum with
different parameter space. The variation with $T_{ph0}$, $\Gamma$, $R_{ph0}$
and $\Delta t$ are plotted in the top-left, top-right, bottom-left and bottom-
right panels respectively, and color-mapped on the right axis of each panel.
The grey-dashed line is the spectral width. The spectrum is scaled with the
value at peak energy to illustrate the shift in peak energy (direction of the
arrow) with an increase in model parameters.  Figure 5: The plot represents
the variation of $\mathcal{W}$ of time-averaged spectra
($\nu\,\mathscr{F}_{\nu}$) with $\alpha$, for different parameter space. The
variation with $T_{ph0}$, $\Gamma$, $R_{ph0}$ and $\Delta t$ are plotted in
the top-left, top-right, bottom-left and bottom-right panels respectively, and
color-mapped on the right axis of each panel. The black-dot-dashed line
corresponds to $\mathcal{W}$ of the black body spectrum. In the first panel,
the red-dashed line corresponds to $\mathcal{W}$ of the instantaneous
($\nu\,F_{\nu}$) spectrum.

##  3 DISCUSSION AND SUMMARY

The prominence of thermal emission in the GRB prompt phase has been well-
studied by various authors (Ryde, 2004, 2005; Pe’er, 2008; Lundman et al.,
2013; Pe’er & Ryde, 2011) and often supported by observations such as GRB
100724B: (Guiriec et al., 2013), GRB 101219B:(Ghirlanda et al., 2013), GRB
081221: (Hou et al., 2018), GRB 220426A: (Wang et al., 2022). Nevertheless,
the prompt emission also carries a significant contribution of non-thermal
spectral component (Pe’Er et al., 2012; Iyyani et al., 2013; Zhang et al.,
2016; Wang et al., 2023). The evolution of the fireball during its coasting
phase can deviate the spectral shape of the photospheric thermal emission from
the Planck function. In this work, we show the spectral shape and the peak of
the emission from a relativistically expanding fireball are primarily governed
by four quantities namely, $R_{ph0}$, $T_{ph0}$, $\Gamma$ and $\alpha$ (§2.2).
Besides these, the integration time of the dynamically evolving fireball can
extend the low energy end of the modified thermal emission.

The modified thermal emission obtained from an evolving fireball has a strong
dependence on the index $\alpha$. For lower values of $\alpha$ ($\sim$ 0.1),
the spectrum peaks at the high energy corresponding to the photospheric
temperature. On the other hand, for high values of $\alpha$ ($\gtrsim$ 0.4),
it will be governed by the cooled photon temperature encountered during its
evolution (Figure 3). This feature leads to a broad range of spectral widths
starting from $\sim 0.54$ (nearly equal to that of the Planck function) to
$\sim 1.15$. Consistently, the observed $\mathcal{W}$ can be extrapolated to
infer the energetics of the burst ($\eta\ll 1$) (Piran et al., 1993; Axelsson
& Borgonovo, 2015). The model can also be applied to individual bursts to
estimate the contribution of the thermal component and can probe the
energetics of the burst.

The choice of other parameters $R_{ph0}$, $T_{ph0}$ and $\Gamma$ will affect
only in shifting the spectral peak but have minimal impact on the spectral
width (Figure 4 and 5). For a given $\alpha$ ($\gtrsim 0.2$), the
$\mathcal{W}$ will increase with $\Delta t$ only till a certain value. Beyond
this, the evolution will modify the flux below the half maximum and hence
$\mathcal{W}$ will remain constant (In bottom-right panel, figure 5). Axelsson
& Borgonovo (2015) showed that the distribution of $\mathcal{W}$ for a large
sample of GRBs, suggests the most probable width to be $\sim 1$. Incidentally,
this width closely matches the width of the single particle emissivity of the
synchrotron process. Hence, the authors concluded the GRBs with $\mathcal{W}$
¿ 1 to be dominated by the synchrotron process; while the ones with
$\mathcal{W}$ ¡ 1 should be consistent with the modified thermal process.
However, Burgess (2019) showed the synchrotron process itself is capable to
explain many GRBs irrespective of their $\mathcal{W}$ estimated from the Band
function. In our work, we show that the modified thermal spectrum obtained
from the evolving fireball is capable of explaining the spectral widths over
the range $0.54\lesssim\mathcal{W}\lesssim 1.2$.

If the decoupling of photons happens well within the radiation-dominated phase
($\eta>1$), the temperature remains nearly constant whereas $\Gamma$ increases
with the evolving radius of the fireball. This scenario is not considered in
the present work and has the potential to produce a spectrum which differs
from the Planck function. Though the temperature remains constant, the
enhancement in $\Gamma$ will shift the emission to high energies resulting in
a modified thermal spectrum. Interestingly, the evolved photon temperatures
measured by the observer will be hotter than the photospheric temperature in
such a case. A detailed study under this scenario will be presented elsewhere.
In this work, we have considered only the thermal contribution of the
fireball. Besides this, the burst spectra may involve the non-thermal emission
arising from the shock accelerated particle distribution. The later
contribution is evident from the observed polarisation of the GRBs
(Chattopadhyay et al., 2021; Kole et al., 2020). Nevertheless, the observed
peak spectral width suggests the thermal origin to be the dominant emission
process.

The authors acknowledge the referee for the valuable suggestions. The authors
also like to thank Dipankar Bhattacharya and A.R. Rao for the insightful
discussions.

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