On the complexity of finding a sun in a graph

TL;DR

This paper proves that determining whether a general graph contains a sun is NP-complete, contrasting with polynomial-time solutions known for specific graph classes like chordal and HHD-free graphs.

Contribution

It establishes the NP-completeness of the sun detection problem in arbitrary graphs, extending the understanding of its computational complexity.

Findings

Sun detection is polynomial-time in chordal graphs.

Sun detection is polynomial-time in HHD-free graphs.

Deciding sun presence in arbitrary graphs is NP-complete.

Abstract

The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it does not contain an induced cycle of length at least four. A graph is strongly chordal if it is chordal and every even cycle has a chord joining vertices whose distance on the cycle is odd. Farber proved that a graph is strongly chordal if and only if it is chordal and contains no induced suns. There are well known polynomial-time algorithms for recognizing a sun in a chordal graph. Recently, polynomial-time algorithms for finding a sun for a larger class of graphs, the so-called HHD-free graphs, have been discovered. In this paper, we prove the problem of deciding whether an arbitrary graph contains a sun in NP-complete.