An Upper Bound for the Double Domination Number in Maximal Outerplanar Graphs
For graph theorists studying domination parameters, this paper corrects and validates a previously proposed bound, but the result is incremental.
The paper provides a complete proof for an upper bound on the double domination number in maximal outerplanar graphs, correcting an incomplete proof from a previous work. The bound is (n+k)/2, where k is the number of pairs of consecutive vertices on the outer cycle at distance at least 3.
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.