Transactions on Combinatorics (Dec 2014)
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.
