Dominating sets inducing large components in maximal outerplanar graphs

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 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 H k has a dominating set D of order at most ⌊