Connected Domination in Plane Triangulations

Abstract

A set of vertices of a graph $G$ such that each vertex of $G$ is either in the set or is adjacent to a vertex in the set is called a dominating set of $G$. If additionally, the set of vertices induces a connected subgraph of $G$ then the set is a connected dominating set of $G$. The domination number $γ(G)$ of $G$ is the smallest number of vertices in a dominating set of $G$, and the connected domination number $γ_c(G)$ of $G$ is the smallest number of vertices in a connected dominating set of $G$. We find the connected domination numbers for all triangulations of up to thirteen vertices. For $n\ge 15$, $n\equiv 0$ (mod 3), we find graphs of order $n$ and $γ_c=\frac{n}{3}$. We also show that the difference $γ_c(G)-γ(G)$ can be arbitrarily large.

Figures (10)

• Figure 1: Left: vertices $v, u, w$, and $t$ have a common neighbor $a$; Right: vertices $v, u, w$ have a common neighbor $a$, vertices $v$ and $t$ have a common neighbor $b$.
• Figure 2: Left: Subgraph of $G$. Right: The set $\{a,b,c\}$ is a connected dominating set of $G$ .
• Figure 3: Graph of order 14 with $\gamma_c=4$.
• Figure 4: Constructing larger triangulations from smaller ones.
• Figure 5: Triangulation of order $n+3$ obtained from a triangulation of order $n$.

Theorems & Definitions (24)

• Lemma 1
• proof
• Corollary 2
• Corollary 3
• Lemma 4
• proof
• Remark 5
• Lemma 6
• proof
• Proposition 7