Domination in Planar Graphs with Small Diameter \(II\)

Abstract

MacGillivray and Seyffarth (J. Graph Theory \(22 (1996),213-229)\) proved that planar graphs of diameter three have domination number at most ten. Recently it was shown (J. Graph Theory \(40 (2002), 1-25)\) that a planar graph of diameter three and of radius two has domination number at most six while every sufficiently large planer graph of diameter three has domination number at most seven. In this paper we improve on these results. We prove that every planar graph of diameter three and of radius two has total domination number (and therefore domination number) at most five. We show then that every sufficiently large planar graph of diameter three has domination number at most six and this result is sharp, while a planar graph of diameter three has domination number at most nine.