!-cycle Compatible Splitting Signed Graphs S(S) and !(S)

A signed graph (or, in short, sigraph) S = (Su,!) consists of an underlying graph Su := G = (V, E) and a function ! : E(Su) −$ {+,−}, called the signature of S. A marking of S is a function μ : V(S) −$ {+,−}. The canonical marking of a signed graph S, denoted μ!, is given as μ!(v) := ! vw%E(S) !(vw). The splitting signed graph S(S) of a signed graph S is formed as follows: • Take a copy of S and for each vertex v of S, take a new vertex v&. Join v& to all vertices u % N(v) by negative edge, if μ!(u) = μ!(v) = − in S and by positive edge otherwise. The splitting signed graph !(S) of a signed graph S is formed as follows: • Take a copy of S and for each vertex v of S, take a new vertex v&. Join v& to all vertices u % N(v) and assign !(uv) as its sign. Here, N(v) is the set of all adjacent vertices to v. A signed graph is called canonically consistent (or !-consistent) if its every cycle contains even number of negative vertices with respect to its canonical marking. A marked signed graph S is called cyclecompatible if for every cycle Z in S, the product of signs of its vertices equals the product of signs of its edges. A signed graph S is !-cycle compatible if for every cycle Z in S, ! e%E(Z) !(e) = ! v%V(Z) μ!(v). In this paper, we establish a structural characterization of signed graph S for which S(S) and !(S) are isomorphic and !-cycle compatible.