Turning cliques into paths to achieve planarity

Motivated by hybrid graph representations, we introduce and study the following beyond-planarity problem, which we call hClique2Path 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.