An Upper Bound for the Double Domination Number 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.

Figures (15)

• Figure 1: The situation that was overlooked in the proof.
• Figure 2: Lemma \ref{['lem:dist']}. Shaded triangles are internal triangles.
• Figure 3: Claim \ref{['clm:2']}. Possible triangles adjacent to $F_{3}$. Shaded triangles are internal triangles.
• Figure 4: Claim \ref{['clm:4']}. Possible triangles adjacent to $F_{4}$ when (a) $F_{5} = \{u_{5},u_{6},u_{7}\}$ and (b) $F_{5} = \{u_{4},u_{6},u_{7}\}$.
• Figure 5: Claim \ref{['clm:5']}. Possible triangles adjacent to $F_{5}$ when (a) $F_{6} = \{u_{6},u_{7},u_{8}\}$ and (b) $F_{5} = \{u_{4},u_{7},u_{8}\}$.

Theorems & Definitions (17)

• Theorem 1.1: zhuang22:_doubl
• Theorem 1.2: zhuang22:_doubl
• Theorem 2.1: Theorem 2.1 in aziz22
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Claim 3.4
• proof : Proof of Claim \ref{['clm:2']}
• Claim 3.5