Clique Separator Decomposition of Hole- and Diamond-Free Graphs and Algorithmic Consequences

TL;DR

This paper characterizes the structure of atoms in hole- and diamond-free graphs, showing they are either weakly chordal or have a specific form, enabling efficient algorithms for related problems.

Contribution

It provides a structural characterization of atoms in hole- and diamond-free graphs, leading to improved algorithmic solutions.

Findings

Atoms of hole- and diamond-free graphs are either weakly chordal or have a specific structure.

Hole- and paraglider-free graphs are perfect graphs.

The structural insights enable efficient algorithms for various problems.

Abstract

Clique separator decomposition introduced by Tarjan and Whitesides is one of the most important graph decompositions. A graph is an {\em atom} if it has no clique separator. A {\em hole} is a chordless cycle with at least five vertices, and an {\em antihole} is the complement graph of a hole. A graph is {\em weakly chordal} if it is hole- and antihole-free. $K_4-e$ is also called {\em diamond}. {\em Paraglider} has five vertices four of which induce a diamond, and the fifth vertex sees exactly the two vertices of degree two in the diamond. In this paper we show that atoms of hole- and diamond-free graphs (of hole- and paraglider-free graphs, respectively) are either weakly chordal or of a very specific structure. Hole- and paraglider-free graphs are perfect graphs. The structure of their atoms leads to efficient algorithms for various problems.