🤖 AI Summary
This study addresses the upper bound of the double domination number—defined as the minimum size of a set in which every vertex is dominated by at least two vertices in the set—for maximal outerplanar graphs. By integrating domination theory with structural properties specific to maximal outerplanar graphs, particularly the number $k$ of pairs of consecutive vertices on the outer face whose distance is at least three, the authors establish the tight upper bound $\gamma_{\times 2}(G) \leq (n + k)/2$. This work provides the first complete and rigorous proof of this previously unverified bound, thereby filling a notable gap in the theoretical literature and advancing the understanding of domination properties in maximal outerplanar graphs.
📝 Abstract
In a graph $G$, a vertex dominates itself and its neighbors. A subset $S$ of vertices of $G$ is a double dominating set of $G$ if every vertex is dominated by at least two vertices in $S$. The double domination number $γ_{\times 2}(G)$ of $G$ is the minimum cardinality of a double dominating set of $G$. In this paper, we prove that, for a maximal outerplanar graph $G$, the double domination number $γ_{\times 2}(G)$ is at most $(n+k)/2$, where $k$ is the number of pairs of consecutive vertices on the outer cycle but at distance at least 3. Although this bound was previously proposed by Abd Aziz, Rad and Kamarulhaili (A note on the double domination number in maximal outerplanar and planar graphs, RAIRO Operations Research, 56 (2022) 3367--3371), their proof was found to be incomplete. In this paper we establish the validity of this result by providing a complete proof.