Weak {2}-domination number of Cartesian products of cycles

Zepeng Li (), Zehui Shao and Jin Xu
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Zepeng Li: Lanzhou University
Zehui Shao: Chengdu University
Jin Xu: Peking University

Journal of Combinatorial Optimization, 2018, vol. 35, issue 1, No 7, 75-85

Abstract: Abstract For a graph $$G=(V, E)$$ G = ( V , E ) , a weak $$\{2\}$$ { 2 } -dominating function $$f:V\rightarrow \{0,1,2\}$$ f : V → { 0 , 1 , 2 } has the property that $$\sum _{u\in N(v)}f(u)\ge 2$$ ∑ u ∈ N ( v ) f ( u ) ≥ 2 for every vertex $$v\in V$$ v ∈ V with $$f(v)= 0$$ f ( v ) = 0 , where N(v) is the set of neighbors of v in G. The weight of a weak $$\{2\}$$ { 2 } -dominating function f is the sum $$\sum _{v\in V}f(v)$$ ∑ v ∈ V f ( v ) and the minimum weight of a weak $$\{2\}$$ { 2 } -dominating function is the weak $$\{2\}$$ { 2 } -domination number. In this paper, we introduce a discharging approach and provide a short proof for the lower bound of the weak $$\{2\}$$ { 2 } -domination number of $$C_n \Box C_5$$ C n □ C 5 , which was obtained by Stȩpień, et al. (Discrete Appl Math 170:113–116, 2014). Moreover, we obtain the weak $$\{2\}$$ { 2 } -domination numbers of $$C_n \Box C_3$$ C n □ C 3 and $$C_n \Box C_4$$ C n □ C 4 .

Keywords: Roman domination; Rainbow domination; Weak {2}-domination; Cartesian product graph (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (7)

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DOI: 10.1007/s10878-017-0157-6

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