Simultaneous PQ-Ordering with Applications to Constrained Embedding Problems

TL;DR

This paper introduces the new problem of Simultaneous PQ-Ordering, analyzes its computational complexity, and applies it to solve various constrained embedding problems efficiently, including planarity and interval graph recognition.

Contribution

It defines Simultaneous PQ-Ordering, proves NP-completeness in general, identifies a tractable subclass, and develops efficient algorithms for several graph embedding and recognition problems.

Findings

NP-complete in general

Linear-time algorithm for Partially PQ-Constrained Planarity

Quadratic-time algorithm for Simultaneous Embedding with Fixed Edges

Abstract

In this paper, we define and study the new problem Simultaneous PQ-Ordering. Its input consists of a set of PQ-trees, which represent sets of circular orders of their leaves, together with a set of child-parent relations between these PQ-trees, such that the leaves of the child form a subset of the leaves of the parent. Simultaneous PQ-Ordering asks whether orders of the leaves of each of the trees can be chosen simultaneously, that is, for every child-parent relation the order chosen for the parent is an extension of the order chosen for the child. We show that Simultaneous PQ-Ordering is NP-complete in general and that it is efficiently solvable for a special subset of instances, the 2-fixed instances. We then show that several constrained embedding problems can be formulated as such 2-fixed instances. In particular, we obtain a linear-time algorithm for Partially PQ-Constrained Planarity for biconnected graphs, a common generalization of two recently considered embedding problems, and a quadratic-time algorithm for Simultaneous Embedding with Fixed Edges for biconnected graphs with a connected intersection; formerly only the much more restricted case that the intersection is biconnected was known to be efficiently solvable. Both results can be extended to the case where the input graphs are not necessarily biconnected but have the property that each cutvertex is contained in at most two non-trivial blocks. This includes for example the case where both graphs have maximum degree 5. Moreover, we give an optimal linear-time algorithm for recognition of simultaneous interval graphs, improving upon a recent O(n^2 log n)-time algorithm due to Jampani and Lubiw and show that this can be used to also solve the problem of extending partial interval representations of graphs with n vertices and m edges in time O(n + m), improving a recent result of Klav\'ik et al.