Communications in Combinatorics and Optimization (Jan 2018)
Mixed Roman domination and 2-independence in trees
Abstract
Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. A {\em mixed Roman dominating function} (MRDF) of $G$ is a function $f:V\cup E\rightarrow \{0,1,2\}$ satisfying the condition that every element $x\in V\cup E$ for which $f(x)=0$ is adjacent or incident to at least one element $y\in V\cup E$ for which $f(y)=2$. The weight of an MRDF $f$ is $\sum _{x\in V\cup E} f(x)$. The mixed Roman domination number $\gamma^*_R(G)$ of $G$ is the minimum weight among all mixed Roman dominating functions of $G$. A subset $S$ of $V$ is a 2-independent set of $G$ if every vertex of $S$ has at most one neighbor in $S$. The maximum cardinality of a 2-independent set of $G$ is the 2-independence number $\beta_2(G)$. These two parameters are incomparable in general, however, we show that if $T$ is a tree, then $\frac{4}{3}\beta_2(T)\ge \gamma^*_R(T)$. Moreover, we characterize all trees attaining the equality.
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