then the Euclidean distance between $b$ and $y$ must be greater than one unit, contradicting the unit interval property.

Every unit interval graph is (claw,net)-free.

When a Hamiltonian path is found, Algorithm \[alg:intervalgraph\] can use the induced path as input, and assign positions to the vertices in linear time.

\[thm:hamiltonian\] Every (claw,net)-free graph $G$ has a Hamiltonian path and such a path can be found in $\OO(n + m)$ time.

By Theorem \[thm:hamiltonian\], we can use the algorithm given in [@linearhamiltonian], and find a Hamiltonian path in a collinear group. Then, we can simply give the found path as the input of Algorithm \[alg:intervalgraph\], and obtain the relative positions of vertices in a collinear group in $\OO(m + n)$ running time.

Localizing collinear groups with respect to each other
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We have proposed a method to localize the vertices of a collinear group in 1D. Now, we describe an algorithm to find the position of two hyperplanar groups in their ambient space. This algorithm is described in general for $\mathbb{R}^d$, and therefore is applicable to both 2D and 3D.

It fixes the position of $d$ vertices in a hyperplanar group, and then uses these positions to determine the equation of 