NodeTrix Planarity Testing with Small Clusters

TL;DR

This paper investigates the computational complexity of NodeTrix planarity testing for flat clustered graphs with small cluster sizes, providing algorithms for fixed sides and establishing NP-completeness results for larger clusters.

Contribution

It introduces algorithms for NodeTrix planarity testing with fixed sides for graphs reducible to partial 2-trees and proves NP-completeness for larger cluster sizes.

Findings

Polynomial-time algorithm for fixed sides when k=2 on partial 2-trees

NP-completeness for k>2 in fixed sides model

NP-completeness for k>4 in free sides model

Abstract

We study the NodeTrix planarity testing problem for flat clustered graphs when the maximum size of each cluster is bounded by a constant $k$. We consider both the case when the sides of the matrices to which the edges are incident are fixed and the case when they can be chosen arbitrarily. We show that NodeTrix planarity testing with fixed sides can be solved in $O(k^{3k+\frac{3}{2}} \cdot n)$ time for every flat clustered graph that can be reduced to a partial 2-tree by collapsing its clusters into single vertices. In the general case, NodeTrix planarity testing with fixed sides can be solved in $O(n)$ time for $k = 2$, but it is NP-complete for any $k > 2$. NodeTrix planarity testing remains NP-complete also in the free sides model when $k > 4$.