Ordering without Forbidden Patterns

Let F be a set of ordered patterns, i.e., graphs whose vertices are linearly ordered. An F-free ordering of the vertices of a graph H is a linear ordering of V (H) such that none of the patterns in F occurs as an induced ordered subgraph. We denote by Ord(F) the decision problem asking whether an input graph admits an F-free ordering; we also use Ord(F) to denote the class of graphs that do admit an F-free ordering. It was observed by Damaschke (and others) that many natural graph classes can be described as Ord(F) for sets F of small patterns (with three or four vertices). This includes bipartite graphs, 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 F consisting of patterns with three vertices, Ord(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, i.e., we solve in polynomial time the problem Ord3 in which the input is a set F of patterns with at most three vertices and a graph H, and the problem is to decide whether or not H admits an Ffree 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 Ord(F) with 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 Ord(F) with F consisting of larger patterns have been shown to be NP-complete by Duffus, Ginn, and Rödl, and we add two simple examples. We also discuss a bipartite version of the problem, BiOrd(F), in which the input is a bipartite graph H with a fixed bipartition of the vertices, and we are given a set F of bipartite patterns. We give a unified polynomial-time algorithm for all problems BiOrd(F) where F has at most four vertices, i.e., we solve the analogous problem BiOrd4. This is also a certifying algorithm, and it yields a unified forbidden substructure characterization for all bipartite graph classes described by some BiOrd(F) with 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 F of forbidden patterns, Ord(F) is either polynomial or NP-complete. ? supported by NSERC Canada 1 Problem definition and motivation For every positive integer k we write [k] = {1, 2, . . . , k}, Ek = {{i, j} | i, j ∈ [k], i 6= j}, and Fk = 2k . Each element in Fk can be viewed as a labelled graph on vertex set [k] and is called a pattern of order k, or simply a k-pattern. Given an input graph H and a linear ordering < of its vertices, we say that a pattern F ∈ Fk occurs in H (under the ordering <) if H contains vertices v1 < v2 < · · · < vk such that the induced ordered subgraph on these vertices is isomorphic to F , i.e., for every i, j ∈ [k], vivj ∈ E(H) if and only if {i, j} ∈ F . For convenience, we shall henceforth write ij to simplify notation for unordered pairs {i, j}. For a set F ⊆ Fk we say that a linear ordering < of V (H) is F-free if none of the patterns in F occurs in <. The problem Ord(F) asks whether or not the input graph H has an F-free ordering. We also consider the problem Ordk that asks, for an input F ⊆ Fk and a graph H, whether or not H has an F-free ordering. The problems Ord(F) can be viewed as 2-satisfiability problems with additional ordering constraints, or as special ternary constraint satisfaction problems, but there are no general algorithms known for such problems [15]. The problems Ord(F) have been studied by Damaschke [5], Duffus, Ginn, and Rödl [6], and others. In particular, Damaschke lists many graph classes that can be equivalently described as Ord(F). For example [2], it is well known that a graph H is chordal if and only if it admits an F-free ordering for F consisting of the single pattern {12, 13}, and H is an interval graph if and only if it admits an F-free ordering for F consisting of the pattern {{13}, {13, 23}}. Similar sets of patterns from F3 describe proper interval graphs, bipartite graphs, split graphs, and comparability graphs [5]. With patterns from F4 we can additionally describe strongly chordal graphs [7], circular-arc graphs [21], and many other graph classes. Analogous definitions apply to bipartite graphs: a bipartite pattern of order k is a bipartite graph whose vertices in each part of the bipartition are labelled by elements of [`] respectively [`′], with ` + `′ ≤ k. We again denote by Bk the set of all bipartite patterns of order k. The problem BiOrd(F) for a fixed F ⊆ Bk asks whether or not an input bipartite graph H with a given bipartition V (H) = U ∪ V admits an ordering of U and of V so that no pattern from F occurs. We also define the corresponding problem BiOrdk in which both F ⊆ Bk and H with V (H) = U ∪ V are part of the input. Several known bipartite graph classes can be characterized as BiOrd(F) for F ⊆ B4. For instance, for F = {11′, 31′} (here ` = 3, `′ = 1), the class BiOrd(F) consists precisely of convex bipartite graphs, and F = {{11′, 12′, 21′}, {12′, 21′}, {12′, 21′, 22′}} (here ` = `′ = 2) similarly defines bipartite permutation graphs (a.k.a., proper interval bigraphs) [14, 24, 25]. One can similarly obtain the classes of chordal bipartite graphs, and bipartite co-circular arc bigraphs [17].