On $bullet$-lict signed graphs $L_{bullet_c}(S)$ and $bullet$-line signed graphs $L_bullet(S)$

• Main Authors: Mukti Acharya, Rashmi Jain, Sangita Kansal
• Format: Article
• Language: English
• Published: University of Isfahan 2016-03-01
• Series: Transactions on Combinatorics
• Subjects: Signed graph; Canonical marking; $bullet$-line signed graph; $bullet$-lict signed graph
• Online Access: http://www.combinatorics.ir/article_7890_ce0590d708f808d60b29942f13824a78.pdf

A emph{signed graph} (or, in short, emph{sigraph}) $S=(S^u,sigma)$ consists of an underlying graph $S^u :=G=(V,E)$ and a function $sigma:E(S^u)longrightarrow {+,-}$, called the signature of $S$. A emph{marking} of $S$ is a function $mu:V(S)longrightarrow {+,-}$. The emph{canonical marking} of a signed graph $S$, denoted $mu_sigma$, is given as $$mu_sigma(v) := prod_{vwin E(S)}sigma(vw).$$The line-cut graph (or, in short, emph{lict graph}) of a graph $G=(V,E)$, denoted by $L_c(G)$, is the graph with vertex set $E(G)cup C(G)$, where $C(G)$ is the set of cut-vertices of $G$, in which two vertices are adjacent if and only if they correspond to adjacent edges of $G$ or one vertex corresponds to an edge $e$ of $G$ and the other vertex corresponds to a cut-vertex $c$ of $G$ such that $e$ is incident with $c$.In this paper, we introduce emph{Dot-lict signed graph} (or emph{$bullet$-lict signed graph}) $L_{bullet_c}(S)$, which has $L_c(S^u)$ as its underlying graph. Every edge $uv$ in $L_{bullet_c}(S)$ has the sign $mu_sigma(p)$, if $u, v in E(S)$ and $pin V(S)$ is a common vertex of these edges, and it has the sign $mu_sigma(v)$, if $uin E(S)$ and $vin C(S)$.we characterize signed graphs on $K_p$, $pgeq2$, on cycle $C_n$ and on $K_{m,n}$ which are $bullet$-lict signed graphs or $bullet$-line signed graphs, characterize signed graphs $S$ so that $L_{bullet_c}(S)$ and $L_bullet(S)$ are balanced. We also establish the characterization of signed graphs $S$ for which $Ssim L_{bullet_c}(S)$, $Ssim L_bullet(S)$, $eta(S)sim L_{bullet_c}(S)$ and $eta(S)sim L_bullet(S)$, here $eta(S)$ is negation of $S$ and $sim$ stands for switching equivalence.