On Lict sigraphs

A signed graph (marked graph) is an ordered pair S=(G,\sigma)‎ ‎(S=(G,\mu))‎, ‎where G=(V,E) is a graph called the underlying‎ ‎graph of S and \sigma:E\rightarrow\{+,-\}‎ ‎(\mu:V\rightarrow\{+,-\}) is a function‎. ‎For a graph G‎, ‎V(G)‎, ‎E(G) and C(G) denote its vertex set‎, ‎edge set and cut-vertex‎ ‎set‎, ‎respectively‎. ‎The lict graph L_{c}(G) of a graph G=(V,E)‎ ‎is defined as the graph having vertex set  E(G)\cup C(G) in which‎ ‎two vertices are adjacent if and only if they correspond to‎ ‎adjacent edges of G or one corresponds to an edge e_{i} of G‎ ‎and the other corresponds to a cut-vertex c_{j} of G such that‎ ‎e_{i} is incident with c_{j}‎. ‎In this paper‎, ‎we introduce lict‎ ‎sigraphs‎, ‎as a natural extension of the notion of lict graph to‎ ‎the realm of signed graphs‎. ‎We show that every lict sigraph is‎ ‎balanced‎. ‎We characterize signed graphs S and S^{'} for which‎ ‎S\sim L_{c}(S)‎, ‎\eta(S)\sim L_{c}(S)‎, ‎L(S)\sim L_{c}(S')‎, ‎J(S)\sim L_{c}(S^{'}) and T_{۱}(S)\sim L_{c}(S^{'})‎, ‎where‎ ‎\eta(S)‎, ‎L(S)‎, ‎J(S) and T_{۱}(S) are negation‎, ‎line graph‎, ‎jump graph and semitotal line sigraph of S‎, ‎respectively‎, ‎and‎ ‎\sim means switching equivalence‎.

Keywords:

signed graph , Line sigraph , Jump sigraph , Semitotal line sigraph , Lict sigraph