On the structure of (pan, even hole)-free graphs

TL;DR

This paper characterizes the structure of (pan, even hole)-free graphs, showing they can be decomposed into circular-arc graphs, leading to efficient algorithms for recognition and coloring, and bounds on their tree-width and chromatic number.

Contribution

It provides a new structure theorem for (pan, even hole)-free graphs, enabling efficient recognition, coloring algorithms, and bounds on graph parameters.

Findings

Decomposition into circular-arc graphs via clique cutsets.

Efficient certifying recognition algorithm with O(nm) complexity.

Coloring algorithm with O(n^{2.5}+nm) complexity.

Abstract

A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-free graph can be decomposed by clique cutsets into essentially unit circular-arc graphs. This structure theorem is the basis of our $O(nm)$-time certifying algorithm for recognizing (pan, even hole)-free graphs and for our $O(n^{2.5}+nm)$-time algorithm to optimally color them. Using this structure theorem, we show that the tree-width of a (pan, even hole)-free graph is at most 1.5 times the clique number minus 1, and thus the chromatic number is at most 1.5 times the clique number.