Three conjectures on the signed cycle domination in graphs

Jian Guan, Xiaoyan Liu, Changhong Lu () and Zhengke Miao
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Jian Guan: Central South University
Xiaoyan Liu: East China Normal University
Changhong Lu: East China Normal University
Zhengke Miao: Jiangsu Normal University

Journal of Combinatorial Optimization, 2013, vol. 25, issue 4, No 12, 639-645

Abstract: Abstract Let G=(V,E) be a graph, a function g:E→{−1,1} is said to be a signed cycle dominating function (SCDF for short) of G if ∑ e∈E(C) g(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as γ sc (G)=min{∑ e∈E(G) g(e)∣g is an SCDF of G}. Xu (Discrete Math. 309:1007–1012, 2009) first researched the signed cycle domination number of graphs and raised the following conjectures: (1) Let G be a maximal planar graphs of order n≥3. Then γ sc (G)=n−2; (2) For any graph G with δ(G)=3, γ sc (G)≥1; (3) For any 2-connected graph G, γ sc (G)≥1. In this paper, we present some results about these conjectures.

Keywords: Domination number; Signed cycle domination number; Planar graph; Maximal planar graph (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10878-012-9506-7

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