(k,p)-Planarity: A Relaxation of Hybrid Planarity
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We present a new model for hybrid planarity that relaxes existing hybrid representations. A graph G = (V,E) is (k,p)-planar if V can be partitioned into clusters of size at most k such that G admits a drawing where: (i) each cluster is associated with a closed, bounded planar region, called a cluster region; (ii) cluster regions are pairwise disjoint, (iii) each vertex v ∈ V is identified with at most p distinct points, called ports, on the boundary of its cluster region; (iv) each inter-cluster edge (u,v) ∈ E is identified with a Jordan arc connecting a port of u to a port of v; (v) inter-cluster edges do not cross or intersect cluster regions except at their endpoints. We first tightly bound the number of edges in a (k,p)-planar graph with p<k. We then prove that (4,1)-planarity testing and (2,2)-planarity testing are NP-complete problems. Finally, we prove that neither the class of (2,2)-planar graphs nor the class of 1-planar graphs contains the other, indicating that the (k,p)-planar graphs are a large and novel class.
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