Dominating Sets inducing Large Components in Maximal Outerplanar Graphs

TL;DR

This paper generalizes existing bounds on domination numbers in maximal outerplanar graphs by providing a unified proof that relates the size of dominating sets to the structure of the induced subgraph, with specific exceptions.

Contribution

It introduces a unified approach to bounding domination numbers in maximal outerplanar graphs, extending previous results and identifying specific exceptional graph classes.

Findings

For large enough graphs, dominating sets with large components exist outside a finite set of exceptions.

The bounds depend on a parameter k, generalizing previous specific cases.

A unified proof technique is provided for these domination bounds.

Abstract

For a maximal outerplanar graph $G$ of order $n$ at least $3$, Matheson and Tarjan showed that $G$ has domination number at most $n/3$. Similarly, for a maximal outerplanar graph $G$ of order $n$ at least $5$, Dorfling, Hattingh, and Jonck showed, by a completely different approach, that $G$ has total domination number at most $2n/5$ unless $G$ is isomorphic to one of two exceptional graphs of order $12$. We present a unified proof of a common generalization of these two results. For every positive integer $k$, we specify a set ${\cal H}_k$ of graphs of order at least $4k+4$ and at most $4k^2-2k$ such that every maximal outerplanar graph $G$ of order $n$ at least $2k+1$ that does not belong to ${\cal H}_k$ has a dominating set $D$ of order at most $\lfloor\frac{kn}{2k+1}\rfloor$ such that every component of the subgraph $G[D]$ of $G$ induced by $D$ has order at least $k$.