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Published in:
20-03-2019
Extremal digraphs for an upper bound on the Roman domination number
Published in: Journal of Combinatorial Optimization | Issue 3/2019
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Abstract
Let \(D=(V,A)\) be a digraph. A Roman dominating function of a digraph D is a function f :\(V\longrightarrow \{0,1,2\}\) such that every vertex u for which \(f(u)=0\) has an in-neighbor v for which \(f(v)=2\). The weight of a Roman dominating function is the value \(f(V)=\sum _{u\in V}f(u)\). The minimum weight of a Roman dominating function of a digraph D is called the Roman domination number of D, denoted by \(\gamma _{R}(D)\). In this paper, we characterize some special classes of oriented graphs, namely out-regular oriented graphs and tournaments satisfying \(\gamma _{R}(D)=n-\Delta ^{+}(D)+1\). Moreover, we characterize digraphs D for which the equality \(\gamma _{R}(D)+\gamma _{R}(\overline{D})=n+3\) holds, where \(\overline{D}\) is the complement of D. Finally, we prove that the problem of deciding whether an oriented graph D satisfies \(\gamma _{R}(D)=n-\Delta ^{+}(D)+1\) is CO-\(\mathcal {NP}\)-complete.