Domination in planar graphs with small diameter*

Abstract

MacGillivray and Seyffarth (J Graph Theory 22 (1996), 213-229) proved that planar graphs of diameter two have domination number at most three and planar graphs of diameter three have domination number at most ten. They also give examples of planar graphs of diameter four having arbitrarily large domination numbers. In this paper we improve on their results. We prove that there is in fact a unique planar graph of diameter two with domination number three, and all other planar graphs of diameter two have domination number at most two. We also prove that every planar graph of diameter three and of radius two has domination number at most six. We then show that every sufficiently large planar graph of diameter three has domination number at most seven. Analogous results for other surfaces are discussed. ß

On dominating sets of maximal outerplanar and planar graphs

“…In particular, bounds on the domination number of a graph in terms of its order and minimum degree have attracted much attention [12,13,16]. Motivated with the applications of the dominating set problem and Conjectures of domination numbers of planar graphs, bounds on the domination number of planar graphs of small diameter [3,5,10] have received much attention. In addition, Honjo, Kawarabayashi and Nakamoto [8] extended Matheson and Tarjan's bound of n 3 to triangulations of other surfaces.…”

Total Domination in Graphs with Diameter 2

“…Bounds on the total domination number of a graph in terms of its order and minimum degree can be found, for example, in . Goddard and Henning showed that the total domination number of a diameter‐2 planar graph is at most 3. However, there exist diameter‐2 nonplanar graphs G of arbitrarily large order n such that .…”

Power domination in planar graphs with small diameter

“…It was proven in [10] that every planar graph of the diameter three and of the radius two has the domination number at most six and that every sufficiently large planar graph of the diameter three has the domination number at most seven. Recently, Dorfling, et al [8] improved the results presented in [9] and [10].…”

“…It was proven in [10] that every planar graph of the diameter three and of the radius two has the domination number at most six and that every sufficiently large planar graph of the diameter three has the domination number at most seven. Recently, Dorfling, et al [8] improved the results presented in [9] and [10]. For a planar graph G of the diameter three, they proved that G has the domination number at most nine; if rad(G) = 2, then the total domination number of G is at most five; if G is sufficiently large, then G has the domination number six and this bound is sharp.…”

“…Thereafter, it was proven in [10] that every planar graph of the diameter 2 has the domination number at most 2 except for the graph F in Fig.4 which has the domination number 3. Since γ P (G) γ(G), it follows that γ P (G) 2 for a planar graph of the diameter two except for the graph F .…”