A lower bound for secure domination number of an outerplanar graph (Q6611009)

``A subset \(S\) of vertices in a graph \(G\) is a secure dominating set of \(G\) if \(S\) is a dominating set of \(G\) and, for each vertex \(u\not \in S\), there is a vertex \(v \in S\) such that \(u v\) is an edge and \((S \setminus \{v\}) \cup \{u\}\) is also a dominating set of \(G\). The secure domination number of \(G\), denoted by \(\gamma_s (G)\), is the cardinality of the smallest secure dominating sets of \(G\)''.\N\textit{E. J. Cockayne} et al. [Util. Math. 67, 19--32 (2005; Zbl 1081.05083)] introduced the problem of secure domination. We know that a graph \(G\) is outerplanar if it has a crossing-free embedding in the plane such that all vertices belong to the boundary of its outer face. It is called a maximal outerplanar graph if it is an outerplanar graph such that the addition of a single edge results in a graph that is not outerplanar. The author in this paper shows that, for any outerplanar graph with \(n \geq 4\) vertices, \(\gamma_s (G) \geq (n+4) / 5\) and the bound is tight.