A characterization of trees with equal Roman {2}-domination and Roman domination numbers

Gonzalez Yero, Ismael; Cabrera Martinez, Abel

doi:10.22049/cco.2019.26364.1103

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	A characterization of trees with equal Roman {2}-domination and Roman domination numbers
Communications in Combinatorics and Optimization
مقاله 3، دوره 4، شماره 2، اسفند 2019، صفحه 95-107 اصل مقاله (383.97 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2019.26364.1103
نویسندگان
Ismael Gonzalez Yero^* ¹؛ Abel Cabrera Martinez²
¹University of Cadiz
²Universitat Rovira i Virgili
چکیده
Given a graph $G=(V,E)$ and a vertex $v \in V$, by $N(v)$ we represent the open neighbourhood of $v$. Let $f:V\rightarrow \{0,1,2\}$ be a function on $G$. The weight of $f$ is $\omega(f)=\sum_{v\in V}f(v)$ and let $V_i=\{v\in V \colon f(v)=i\}$, for $i=0,1,2$. The function $f$ is said to be 1) a Roman $\{2\}$-dominating function, if for every vertex $v\in V_0$, $\sum_{u\in N(v)}f(u)\geq 2$. The Roman $\{2\}$-domination number, denoted by $\gamma_{\{R2\}}(G)$, is the minimum weight among all Roman $\{2\}$-dominating functions on $G$; 2) a Roman dominating function, if for every vertex $v\in V_0$ there exists $u\in N(v)\cap V_2$. The Roman domination number, denoted by $\gamma_R(G)$, is the minimum weight among all Roman dominating functions on $G$. It is known that for any graph $G$, $\gamma_{\{R2\}}(G)\leq \gamma_R(G)$. In this paper, we characterize the trees $T$ that satisfy the equality above.
کلیدواژه‌ها
Roman ${2}$-domination؛ $2$-rainbow domination؛ Roman domination؛ tree

مراجع
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A characterization of trees with equal Roman {2}-domination and Roman domination numbers