Geodesic
A note on signed cycle domination in graphs
Kragujevac Journal of Mathematics, Tome 37 (2013) no. 1, p. 159

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

Let $G = (V, E)$ be a simple graph. A function $f : E \rightarrow\{-1, 1\}$ is said to be a signed cycle dominating function (SCDF) of $G$ if $um_{eı E(C)}f (e) \ge 1$ holds for every induced cycle $C$ of $G$. The signed cycle domination number of $G$ is defined as $\gamma_{sc}'(G) = \min\{um_{eı E(G)} f (e) \mid f \mbox{ is an } SCDF \mbox{ of } G\}$. B.\ Xu [4] conjectured that for any maximal planar graph $G$ of order $n \ge 3$, $\gamma_{sc}'(G) =n-2$. In this paper, we first prove that the conjecture is true and then we show that if $G$ is a connected cubic claw-free graph of order $n$, then $\gamma_{sc}'(G)\leq n$.
Classification : 05C69
Keywords: Planar graph, signed cycle domination. 
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     title = {A note on signed cycle domination in graphs},
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Hossein Karami; Rana Khoeilar; Seyed Mahmoud Sheikholeslami. A note on signed cycle domination in graphs. Kragujevac Journal of Mathematics, Tome 37 (2013) no. 1, p. 159 . http://geodesic.mathdoc.fr/item/KJM_2013_37_1_a11/