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Negation switching invariant 3-Path signed graphs

Abstract Formally, a signed graph (sigraph) S is a pair (G, σ) that consists of a graph G = (V, E) and a sign mapping or signature σ from E to the sign group {+, –}. Given a sigraph S and a positive integer t, the t-path sigraph (S)t of S is a sigraph whose vertex set is V(S) and two vertices are adjacent if and only if there exist a path of length t between these vertices and then by defining its sign st(e) to be ‘-’ if and only if in every such path of length t in S all the edges are negative. The negation η(S) of a sigraph S is a sigraph obtained from S by reversing the sign of every edge of S. Two sigraphs S1 and S2 on the same underlying graph are switching equivalent if it is possible to assign signs ‘+’ (‘plus’) or ‘-’ (‘minus’) to the vertices of S1 such that by reversing the sign of each of its edges that has received opposite signs at its ends one obtains S2. In this paper, we characterize sigraphs whose negations are switching equivalent to their t-path sigraphs for t = 3.