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We study the Length-Bounded Cut problem for special graph classes and from a parameterized complexity viewpoint. Here, we are given a graph G , two vertices s and t , and positive integers β and λ. The task is to find a set F of at most β edges such that each s - t -path of length at most λ in G contains some edge in F. Bazgan et al. [20] conjectured that Length-Bounded Cut admits a polynomial-time algorithm if the input graph is a proper interval graph. We confirm this conjecture by providing a dynamic-programming-based polynomial-time algorithm. Moreover, we strengthen the W[1]-hardness result of Dvořák and Knop [15] for Length-Bounded Cut parameterized by pathwidth by showing W[1]-hardness for the combined parameter pathwidth and maximum degree of the input graph. Finally, we prove that Length-Bounded Cut is W[1]-hard for the feedback vertex number. Both our hardness results complement known XP algorithms. [ABSTRACT FROM AUTHOR]

Similar to the satisfiability (SAT) problem, which can be seen to be the archetypical problem for NP, the quantified Boolean formula problem (QBF) is the archetypical problem for PSPACE. Recently, Atserias and Oliva (2014) showed that, unlike for SAT, many of the well-known decompositional parameters (such as treewidth and pathwidth) do not allow efficient algorithms for QBF. The main reason for this seems to be the lack of awareness of these parameters towards the dependencies between variables of a QBF formula. In this paper we extend the ordinary pathwidth to the QBF-setting by introducing prefix pathwidth, which takes into account the dependencies between variables in a QBF, and show that it leads to an efficient algorithm for QBF. We hope that our approach will help to initiate the study of novel tailor-made decompositional parameters for QBF and thereby help to lift the success of these decompositional parameters from SAT to QBF. [ABSTRACT FROM AUTHOR]

We consider a bicriterion generalization of the pathwidth problem: given integers k , l and a graph G , does there exist a path decomposition of G of width at most k and length (i.e., number of bags) at most l ? We provide a complete complexity classification of the problem in terms of k and l for general graphs. Contrary to the original pathwidth problem, which is fixed-parameter tractable with respect to k , the generalized problem is NP-complete for any fixed k ≥ 4 , and also for any fixed l ≥ 2 . On the other hand, we give a polynomial-time algorithm that constructs a minimum-length path decomposition of width at most k ≤ 3 for any disconnected input graph. As a by-product, we obtain an almost complete classification for connected graphs: the problem is NP-complete for any fixed k ≥ 5 , and polynomial for any k ≤ 3 . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate algorithms for some related graph parameters. Each of these asks for a linear ordering of the vertices of the graph (or can be formulated as such), and constructive linear time algorithms for the fixed parameter versions of the problems have been published for several of these. Examples are cutwidth, pathwidth, and directed or weighted variants of these. However, these algorithms have complicated technical details. This paper attempts to present ideas in these algorithms in a different more easily accessible manner, by showing that the algorithms can be obtained by a stepwise modification of a trivial hypothetical non-deterministic algorithm. The methodology is applied to rederive known results for the cutwidth and the pathwidth problem, and obtain new results for several variants of these problems, like directed and weighted variants of cutwidth and modified cutwidth. [Copyright &y& Elsevier]

Abstract: In this article we consider the application of ideas from parameterized complexity, and topological graph theory, to online problems. We focus on parameterized promise problems, where we are promised that the problem input obeys certain properties, or is presented in a certain fashion. We explore the effects of using graph width metrics as restrictions on the input to online problems. It seems natural to suppose that, for graphs having some form of bounded width, good online algorithms may exist for a number of natural problems. In the work presented we concentrate on online graph coloring problems, where we restrict the allowed input to instances having some form of bounded treewidth or pathwidth. We also consider the effects of restricting the presentation of the input to some form of bounded width decomposition or layout. A consequence of this part of the work is the clarification of a new parameter for graphs, persistence, which arises naturally in the online setting, and is of interest in its own right. We present some basic results regarding the general recognition of graphs having bounded persistence path decompositions. [Copyright &y& Elsevier]