Finding a Sun in Building-Free Graphs

Abstract

Deciding whether an arbitrary graph contains a sun was recently shown to be NP-complete (Hoang in SIAM J Discret Math 23:2156–2162, 2010). We show that whether a building-free graph contains a sun can be decided in O(min{mn 3, m 1.5 n 2}) time and, if a sun exists, it can be found in the same time bound. The class of building-free graphs contains many interesting classes of perfect graphs such as Meyniel graphs which, in turn, contains classes such as hhd-free graphs, i-triangulated graphs, and parity graphs. Moreover, there are imperfect graphs that are building-free. The class of building-free graphs generalizes several classes of graphs for which an efficient test for the presence of a sun is known. We also present a vertex elimination scheme for the class of (building, gem)-free graphs. The class of (building, gem)-free graphs is a generalization of the class of distance hereditary graphs and a restriction of the class of (building, sun)-free graphs.