Intersection graphs of noncrossing arc-connected sets in the plane

Abstract

Arc-connected sets (Formula presented) are called noncrossing if both A – B and B – A are arc-connected. A graph is called an NCAC graph if it has an intersection representation in which vertices are represented by arc-connected sets in the plane and any two sets of the representation are noncrossing. In particular, disk intersection graphs are NCAC. By a unified reduction we show that recognition of disk intersection and NCAC graphs are NP-hard. A simple observation shows that triangle-free disk intersection and NCAC graphs are planar, and hence recognizable in polynomial time. On the other hand, recognition of triangle-free AC graphs (intersection graphs of arc-connected sets) is still NP-hard.