(kp)-Planarity: A Relaxation of Hybrid Planarity

  • Emilio Di Giacomo
  • William J. Lenhart
  • Giuseppe Liotta
  • Timothy W. RandolphEmail author
  • Alessandra Tappini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)


We present a new model for hybrid planarity that relaxes existing hybrid representations. A graph \(G = (V,E)\) is (kp)-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 \(v \in V\) is identified with at most p distinct points, called ports, on the boundary of its cluster region; (iv) each inter-cluster edge \((u,v) \in E\) 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 (kp)-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 (kp)-planar graphs are a large and novel class.


\((k, p)\)-Planarity Hybrid representations Clustered graphs 


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Emilio Di Giacomo
    • 1
  • William J. Lenhart
    • 2
  • Giuseppe Liotta
    • 1
  • Timothy W. Randolph
    • 3
    Email author
  • Alessandra Tappini
    • 1
  1. 1.Università degli Studi di PerugiaPerugiaItaly
  2. 2.Williams CollegeWilliamstownUSA
  3. 3.Columbia UniversityNew York CityUSA

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