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 $sigma:Erightarrow{+,-}$ $(mu:Vrightarrow{+,-})$ is a function. For a graph $G$, $V(G), E(G)$ and $C(G)$ denote its vertex set, edge set and cut-vertex...
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Format: | Article |
Language: | English |
Published: |
University of Isfahan
2014-12-01
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Series: | Transactions on Combinatorics |
Subjects: | |
Online Access: | http://www.combinatorics.ir/pdf_5627_e7de2aef7c26e21d97bfaf79f2112406.html |
Summary: | 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. |
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ISSN: | 2251-8657 2251-8665 |