Graph Planarity Testing with Hierarchical Embedding Constraints
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Hierarchical embedding constraints define a set of allowed cyclic orders for the edges incident to the vertices of a graph. These constraints are expressed in terms of FPQ-trees. FPQ-trees are a variant of PQ-trees that includes F-nodes in addition to P- and to Q-nodes. An F-node represents a permutation that is fixed, i.e., it cannot be reversed. Let G be a graph such that every vertex of G is equipped with a set of FPQ-trees encoding hierarchical embedding constraints for its incident edges. We study the problem of testing whether G admits a planar embedding such that, for each vertex v of G, the cyclic order of the edges incident to v is described by at least one of the FPQ-trees associated with v. We prove that the problem is NP-complete even when the number of FPQ-trees associated with each vertex is bounded by a constant. If however the branchwidth of G is bounded, the problem can be solved in polynomial time. Besides being interesting on its own right, the study of planarity testing with hierarchical embedding constraints can be used to address other planarity testing problems which can be modeled by associating a set of FPQ-trees to the vertices of the input graph. As a proof of concept, we apply our techniques to the study of NodeTrix planarity testing of clustered graphs. We show that NodeTrix planarity testing with fixed sides is fixed-parameter tractable when parameterized by the size of the clusters and by the tree-width of the multi-graph obtained by collapsing the clusters to single vertices.
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