Graph Planarity Testing with Hierarchical Embedding Constraints

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 fixed-parameter tractable for biconnected graphs, where the parameters are the treewidth of $G$ and the number of FPQ-trees associated with every vertex of $G$. We also show that the problem is NP-complete if parameterized by the number of FPQ-trees only, and W[1]-hard if parameterized by the treewidth only. 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. In particular, 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 treewidth of the multi-graph obtained by collapsing the clusters to single vertices, provided that this graph is biconnected.