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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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/