Disjunctive domination in maximal outerplanar graphs

TL;DR

This work addresses bounding the disjunctive domination number $\gamma_2^d(G)$ for maximal outerplanar graphs (mop). It develops a structural, inductive proof leveraging a tree of triangles and region decompositions, supported by preliminary lemmas and small-order baselines. The main finding is that for any mop of order $n\ge7$ with $k$ degree-2 vertices, $\gamma_2^d(G) \le \left\lfloor \frac{2}{9}(n+k) \right\rfloor$, and this bound is tight. The result sharpens existing domination bounds for mop and provides explicit extremal constructions, contributing to a deeper understanding of disjunctive domination in planar graph classes.

Abstract

A disjunctive dominating set of a graph $G$ is a set $D \subseteq V(G)$ such that every vertex in $V(G)\setminus D$ has a neighbor in $D$ or has at least two vertices in $D$ at distance $2$ from it. The disjunctive domination number of $G$, denoted by $γ_2^d(G)$, is the minimum cardinality of a disjunctive dominating set of $G$. In this paper, we show that if $G$ is a maximal outerplanar graph of order $n \ge 7$ with $k$ vertices of degree $2$, then $γ_2^d(G)\le \lfloor\frac{2}{9}(n+k)\rfloor$, and this bound is sharp.

Figures (40)

• Figure 1: Possible (shaded) triangle adjacent to triangle $F_3$
• Figure 2: Possible (shaded) triangle adjacent to triangle $F_4$ when $V(F_4)=\{u_2,u_5,u_6\}$.
• Figure 3: Possible (shaded) triangle adjacent to triangle $F_4$ when $V(F_4)=\{u_4,u_5,u_6\}$.
• Figure 4: (a) $H_1$, (b) $H_2$, (c) $H_3$, and (d) $H_4$. Possible (shaded) triangle adjacent to triangle $F_5$.
• Figure 5: (a) $H_5$, (b) $H_6$, (c) $H_7$, and (d) $H_8$. Possible (shaded) triangle adjacent to triangle $F_6$.

Theorems & Definitions (80)

• Lemma 1: Ro-83
• Theorem 1
• Lemma 2: CaWa-13
• Lemma 3: Ch-75
• Lemma 4
• proof
• Lemma 5
• proof
• Claim 1
• proof : Proof of \ref{['3-dist']}