Graph Planarity by Replacing Cliques with Paths

Abstract

This paper introduces and studies the following beyond-planarity problem, which we call h-CLIQUE2PATH PLANARITY. Let G be a simple topological graph whose vertices are partitioned into subsets of size at most h, each inducing a clique. h-CLIQUE2PATH PLANARITY asks whether it is possible to obtain a planar subgraph of G by removing edges from each clique so that the subgraph induced by each subset is a path. We investigate the complexity of this problem in relation to k-planarity. In particular, 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 when G is a simple 1-plane graph, for any value of h. Our results contribute to the growing fields of hybrid planarity and of graph drawing beyond planarity.