Counting Independent Sets in Cocomparability Graphs

TL;DR

This paper presents linear-time algorithms for counting independent sets and cliques in cocomparability and comparability graphs, respectively, and explores the computational complexity of related counting problems.

Contribution

The paper introduces the first linear-time algorithms for counting independent sets in cocomparability graphs and extends results to related graph classes and fixed-parameter problems.

Findings

Counting independent sets in cocomparability graphs is linear-time.

Counting cliques in comparability graphs is linear-time.

Counting certain structures in other related graphs is #P-complete.

Abstract

We show that the number of independent sets in cocomparability graphs can be counted in linear time, as can counting cliques in comparability graphs. By contrast, counting cliques in cocomparabilty graphs and counting independent sets in comparability graphs are #P-complete. We extend these results to counting maximal cliques and independent sets. We also consider the fixed-parameter versions of counting cliques and independent sets of given size $k$. Finally, we combine the results to show that both counting cliques and independent sets in permutation graphs are in linear time.