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.
Highlights
A typical problem concerning the visual analysis of real-world networks refers to the creation of occlusions and hairball-like structures in dense subnetworks when node-link diagrams are generated by standard layout algorithms, e.g., force-directed methods
We introduce and study a problem called h-C LIQUE 2PATH P LANARITY, that is a restricted version of C LIQUE - PLANARITY in which the input graph comes with a given embedding and each clique has size at most h
In the version of h-C LIQUE 2PATH P LANARITY that we study, the input graph G is a simple topological graph
Summary
A typical problem concerning the visual analysis of real-world networks refers to the creation of occlusions and hairball-like structures in dense subnetworks when node-link diagrams are generated by standard layout algorithms, e.g., force-directed methods. The NodeTrix model [3] represents a first example of hybrid representation It combines node-link diagrams with adjacency-matrix representations of the denser subgraphs [3,4,5,6]. Clique-planar drawings are defined in [9] as intersection-link representations in which the objects are isothetic rectangles, and the partition into intersection- and link-edges is given as a part of the input, so that the graph induced by the intersection-edges is composed of a set of vertex-disjoint cliques. The corresponding recognition problem, called C LIQUE - PLANARITY, has been proved NP-complete in general and polynomial-time solvable in restricted cases, for example when the rectangle representing each vertex is given as a part of the input, or when the cliques are arranged on levels according to a hierarchy. Preliminary results have been presented in [12]
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