Turning Cliques into Paths to Achieve Planarity
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.
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