Ordering without Forbidden Patterns

Abstract

Let ${\mathcal{F}}$ be a set of ordered patterns, i.e., graphs whose vertices are linearly ordered. An ${\mathcal{F}}$-free ordering of the vertices of a graph H is a linear ordering of V(H) such that none of the patterns in ${\mathcal{F}}$ occurs as an induced ordered subgraph. We denote by $\hbox{\sc Ord}({\mathcal{F}})$ the decision problem asking whether an input graph admits an ${\mathcal{F}}$-free ordering; we also use $\hbox{\sc Ord}({\mathcal{F}})$ to denote the class of graphs that do admit an ${\mathcal{F}}$-free ordering. It was observed by Damaschke (and others) that many natural graph classes can be described as $\hbox{\sc Ord}({\mathcal{F}})$ for sets ${\mathcal{F}}$ of small patterns (with three or four vertices). This includes recognition of split graphs, interval graphs, proper interval graphs, cographs, comparability graphs, chordal graphs, strongly chordal graphs, and so on. Damaschke also noted that for many sets ${\mathcal{F}}$ consisting of patterns with three vertices, $\hbox{\sc Ord}({\mathcal{F}})$ is polynomial-time solvable by known algorithms or their simple modifications. We complete the picture by proving that all these problems can be solved in polynomial time. In fact, we provide a single master algorithm, which solves in polynomial time the problem $\hbox{\sc Ord}_3$ in which the input is a set ${\mathcal{F}}$ of patterns, each with at most three vertices, and a graph H, and the problem is to decide whether or not H admits an ${\mathcal{F}}$-free ordering of the vertices. Our algorithm certifies non-membership by a forbidden substructure, and thus provides a single forbidden structure characterization for all the graph classes described by some $\hbox{\sc Ord}({\mathcal{F}})$ with ${\mathcal{F}}$ consisting of patterns with at most three vertices. This includes bipartite graphs, split graphs, interval graphs, proper interval graphs, chordal graphs, and comparability graphs. Many of the problems $\hbox{\sc Ord}({\mathcal{F}})$ with ${\mathcal{F}}$ consisting of larger patterns have been shown to be NP-complete by Duffus, Ginn, and Rödl, and we add some additional examples.

We also discuss a bipartite version of the problem, $\hbox{\sc BiOrd}({\mathcal{F}})$, in which the input is a bipartite graph H with a fixed bipartition of the vertices, and we are given a set ${\mathcal{F}}$ of bipartite patterns. We give a unified polynomial-time algorithm for all problems $\hbox{\sc BiOrd}({\mathcal{F}})$ where ${\mathcal{F}}$ has at most four vertices, i.e., we solve the analogous problem $\hbox{\sc BiOrd}_4$. This is also a certifying algorithm, and it yields a unified forbidden substructure characterization for all bipartite graph classes described by some $\hbox{\sc BiOrd}({\mathcal{F}})$ with ${\mathcal{F}}$ consisting of bipartite patterns with at most four vertices. This includes chordal bipartite graphs, co-circular-arc bipartite graphs, and bipartite permutation graphs. We also describe some examples of digraph ordering problems and algorithms.

We conjecture that for every set ${\mathcal{F}}$ of forbidden patterns, $\hbox{\sc Ord}({\mathcal{F}})$ is either polynomial or NP-complete.

Supported by NSERC, Canada.