Maximal cliques structure for cocomparability graphs and applications

TL;DR

This paper explores the structure of cocomparability graphs through maximal cliques, introduces a new lattice characterization, and develops efficient algorithms for finding maximal interval subgraphs and simplicial vertices.

Contribution

It provides a novel lattice-based characterization of cocomparability graphs and presents new algorithms with improved efficiency for key problems.

Findings

A lattice structure characterizes cocomparability graphs.

An $O(n+mlogn)$ algorithm finds maximal interval subgraphs.

A linear-time algorithm computes all simplicial vertices.

Abstract

A cocomparability graph is a graph whose complement admits a transitive orientation. An interval graph is the intersection graph of a family of intervals on the real line. In this paper we investigate the relationships between interval and cocomparability graphs. This study is motivated by recent results Corneil,Dalton, Habib (2013) and Dusart, Habib (2016) and that show that for some problems, the algorithm used on interval graphs can also be used with small modifications on cocomparability graphs. Many of these algorithms are based on graph searches that preserve cocomparability orderings. First we propose a characterization of cocomparability graphs via a lattice structure on the set of their maximal cliques. Using this characterization we can prove that every maximal interval subgraph of a cocomparability graph $G$ is also a maximal chordal subgraph of $G$. Although the size of this lattice of maximal cliques can be exponential in the size of the graph, it can be used as a framework to design and prove algorithms on cocomparability graphs. In particular we show that a new graph search, namely Local Maximal Neighborhood Search (LocalMNS) leads to an $O(n+mlogn)$ time algorithm to find a maximal interval subgraph of a cocomparability graph. Similarly we propose a linear time algorithm to compute all simplicial vertices in a cocomparability graph. In both cases we improve on the current state of knowledge.