The complexity of counting poset and permutation patterns

We introduce a notion of pattern occurrence that generalizes both classical permutation patterns as well as poset containment. Many questions about pattern statistics and avoidance generalize naturally to this setting, and we focus on functional complexity problems -- particularly those that arise by constraining the order dimensions of the pattern and text posets. We show that counting the number of induced, injective occurrences among dimension 2 posets is #P-hard; enumerating the linear extensions that occur in realizers of dimension 2 posets can be done in polynomial time, while for unconstrained dimension it is GI-complete; counting not necessarily induced, injective occurrences among dimension 2 posets is #P-hard; counting injective or not necessarily injective occurrences of an arbitrary pattern in a dimension 1 text is #P-hard, although it is in FP if the pattern poset is constrained to have bounded intrinsic width; and counting injective occurrences of a dimension 1 pattern in an arbitrary text is #P-hard, while it is in FP for bounded dimension texts. This framework easily leads to a number of open questions, chief among which are (1) is it #P-hard to count the number of occurrences of a dimension 2 pattern in a dimension 1 text, and (2) is it #P-hard to count the number of texts which avoid a given pattern?Comment: 15 page

Linear Time Recognition of P4-Indifferent Graphs

A simple graph is P4-indifferent if it admits a total order b > c > d. P4-indifferent graphs generalize indifferent graphs and are perfectly orderable. Recently, Hoang,Maray and Noy gave a characterization of P4-indifferent graphs interms of forbidden induced subgraphs. We clarify their proof and describe a linear time algorithm to recognize P4-indifferent graphs. Whenthe input is a P4-indifferent graph, then the algorithm computes an order < as above.Key words: P4-indifference, linear time, recognition, modular decomposition.

A representation for the modules of a graph and applications

We describe a simple representation for the modules of a graph C. We show that the modules of C are in one-to-one correspondence with the ideaIs of certain posets. These posets are characterizaded and shown to be layered posets, that is, transitive closures of bipartite tournaments. Additionaly, we describe applications of the representation. Employing the above correspondence, we present methods for solving the following problems: (i) generate alI modules of C, (ii) count the number of modules of C, (iii) find a maximal module satisfying some hereditary property of C and (iv) find a connected non-trivial module of C

Matching cutsets in graphs of diameter 2

AbstractWe say that a graph has a matching cutset if its vertices can be coloured in red and blue in such a way that there exists at least one vertex coloured in red and at least one vertex coloured in blue, and every vertex has at most one neighbour coloured in the opposite colour. In this paper we study the algorithmic complexity of a problem of recognizing graphs which possess a matching cutset. In particular we present a polynomial-time algorithm which solves this problem for graphs of diameter two

Computing commons interval of K permutations, with applications to modular decomposition of graphs

International audienceWe introduce a new way to compute common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations, that can contain a quadratic number of intervals, as well as a linear space basis of this set of common intervals. Finally, we show how our results on permutations can be used for computing the modular decomposition of graphs in linear time