Graph classes and forbidden patterns on three vertices

TL;DR

This paper systematically characterizes graph classes defined by forbidden patterns on three vertices and proves that most of these classes can be recognized in linear time, advancing understanding of their structure and recognition algorithms.

Contribution

It provides a complete classification of classes defined by three-node forbidden patterns and establishes linear-time recognition algorithms for most of these classes.

Findings

21 out of 23 classes recognized in linear time

Complete characterization of classes defined by three-node patterns

Open questions on the structure of these classes

Abstract

This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.

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