Finding a sun in building-free graphs

Deciding whether an arbitrary graph contains a sun was recently shown to be NP-complete [10]. We show that whether a building-free graph contains a sun can be decided in O(min{mn,mn}) 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 hhdfree graphs, i-triangulated graphs, and parity graphs. Moreover, there are imperfect graphs that are building-free. The class of buildingfree 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.