C-Planarity of Overlapping Clusterings Including Unions of Two Partitions

TL;DR

This paper investigates the computational complexity of clustered planarity with overlapping clusters, providing polynomial-time algorithms for specific cases and proving NP-completeness in more general scenarios.

Contribution

It introduces polynomial-time algorithms for certain overlapping clusterings and establishes NP-completeness for more complex cases, advancing understanding of clustered planarity.

Findings

Polynomial-time solution when each cluster induces a connected subgraph.

Linear-time solution when clusters form the union of two partitions with connected subgraphs.

NP-completeness in restricted cases with multiple components in clusters and their complements.

Abstract

We show that clustered planarity with overlapping clusters as introduced by Didimo et al. can be solved in polynomial time if each cluster induces a connected subgraph. It can be solved in linear time if the set of clusters is the union of two partitions of the vertex set such that, for each cluster, both the cluster and its complement, induce connected subgraphs. Clustered planarity with overlapping clusters is NP-complete, even if restricted to instances where the underlying graph is 2-connected, the set of clusters is the union of two partitions and each cluster contains at most two connected components while their complements contain at most three connected components.