Turning Cliques into Paths to Achieve Planarity

TL;DR

This paper introduces the $h$-Clique2Path Planarity problem, which involves transforming cliques into paths to achieve planarity, and analyzes its computational complexity across different graph classes.

Contribution

It defines a new graph transformation problem motivated by hybrid representations and establishes its NP-completeness and polynomial solvability in specific cases.

Findings

NP-complete for $h=4$ on 3-plane graphs

Linear-time solvable for 1-plane graphs

Links problem complexity to $k$-planarity

Abstract

Motivated by hybrid graph representations, we introduce and study the following beyond-planarity problem, which we call $h$-Clique2Path Planarity: Given a graph $G$, whose vertices are partitioned into subsets of size at most $h$, each inducing a clique, remove edges from each clique so that the subgraph induced by each subset is a path, in such a way that the resulting subgraph of $G$ is planar. We study this problem when $G$ is a simple topological graph, and establish its complexity in relation to $k$-planarity. We prove that $h$-Clique2Path Planarity is NP-complete even when $h=4$ and $G$ is a simple $3$-plane graph, while it can be solved in linear time, for any $h$, when $G$ is $1$-plane.