Intersection Graphs of Noncrossing Arc-Connected Sets in the Plane
Kratochvíl, Jan (1997) Intersection Graphs of Noncrossing Arc-Connected Sets in the Plane. In: Symposium on Graph Drawing, GD '96, September 18-20, 1996, Berkeley, California, , pp. 257-270 (Official URL: http://dx.doi.org/10.1007/3-540-62495-3_53).
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Arc-connected sets A, B \in E_2 are called noncrossing if both A-B, 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.
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