Algorithmic characterization of signed graphs whose two path signed graphs and square graphs are isomorphic

Abstract

A signed graph (or sigraph, in short) is an ordered pair S = (S ^u ,σ), where S ^u is a graph G = (V,E), called the underlying graph of S and σ : E→ {+,-}is a function from the edge set E of S ^u into the set {+,-}, called the signature of S. A sigraph is also denoted as S = (V;E,σ). A two path sigraph (S)2 = (V,E',σ') of a sigraph S = (V,E,σ) is formed by taking a copy of the vertex set V(S) of S, and joining two vertices in copy by a single edge e' = uv whenever there is a u-v path of length two in S and then by defining its sign σ'(e') = - whenever in all the u-v path of length two the edges are negative, positive otherwise. A square sigraph S2 = (V,E',σ) of a sigraph S = (V,E,σ) is defined as a sigraph with V(S ² ) = V(S) with uv adjacent in S ² whenever there exist a path of length two or less in S. And σ'(uv) = σ(uv) if there does not exist a uv path of length two σ'(uv) = - if all the uv paths of length two are negative σ(uv) = + otherwise.