Finding a sun in building-free graphs

TL;DR

This paper presents an efficient algorithm to determine the presence of a sun in building-free graphs, expanding the classes of graphs where such detection is computationally feasible.

Contribution

It introduces a polynomial-time algorithm for detecting suns in building-free graphs and extends the analysis to (building, gem)-free graphs, broadening the scope of efficiently recognizable graph classes.

Findings

Sun detection in building-free graphs is solvable in polynomial time.

Building-free graphs include many perfect graph classes like Meyniel graphs.

The paper provides a vertex elimination scheme for (building, gem)-free graphs.

Abstract

Deciding whether an arbitrary graph contains a sun was recently shown to be NP-complete. We show that whether a building-free graph contains a sun can be decided in O(min$\{m{n^3}, m^{1.5}n^2\}$) 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.