On Lict sigraphs

Abstract

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 $sigma:Erightarrow{+,-}$‎ ‎$(mu:Vrightarrow{+,-})$ 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 ‎$Ssim L_{c}(S)$‎, ‎$eta(S)sim L_{c}(S)$‎, ‎$L(S)sim L_{c}(S')$‎, ‎$J(S)sim L_{c}(S^{'})$ and $T_{1}(S)sim L_{c}(S^{'})$‎, ‎where‎ ‎$eta(S)$‎, ‎$L(S)$‎, ‎$J(S)$ and $T_{1}(S)$ are negation‎, ‎line graph‎, ‎jump graph and semitotal line sigraph of $S$‎, ‎respectively‎, ‎and‎ ‎$sim$ means switching equivalence‎.