Independent sets in edge-clique graphs

TL;DR

This paper investigates the computational complexity of the independent set problem in edge-clique graphs, revealing cases where it is solvable efficiently and others where it remains NP-complete, with implications for graph classes like cographs.

Contribution

It demonstrates the complexity landscape of the independent set problem in edge-clique graphs across various graph classes, including new polynomial-time algorithms and NP-completeness results.

Findings

Edge-clique graphs of cocktail party graphs have unbounded rankwidth.

Independent set problem is solvable in O(n^4) time for cographs and distance-hereditary graphs.

The problem remains NP-complete for graphs without odd wheels.

Abstract

We show that the edge-clique graphs of cocktail party graphs have unbounded rankwidth. This, and other observations lead us to conjecture that the edge-clique cover problem is NP-complete for cographs. We show that the independent set problem on edge-clique graphs of cographs and of distance-hereditary graphs can be solved in O(n^4) time. We show that the independent set problem on edge-clique graphs of graphs without odd wheels remains NP-complete.