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We describe a polynomial-time algorithm which, given a graph G with treewidth t, approximates the path-width of G to within a ratio of O(t√log t). This is the first algorithm to achieve an f(t)-approximation for some function f. Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least th + 2 has treewidth at least t or contains a subdivision of a complete binary tree of height h + 1. The bound th + 2 is best possible up to a multiplicative constant. This result was motivated by, and implies (with c = 2), the following conjecture of Kawarabayashi and Rossman (SODA'18): there exists a universal constant c such that every graph with pathwidth Ω(kc) has treewidth at least k or contains a subdivision of a complete binary tree of height k. Our main technical algorithm takes a graph G and some (not necessarily optimal) tree decomposition of G of width t' in the input, and it computes in polynomial time an integer h, a certificate that G has pathwidth at least h, and a path decomposition of G of width at most (t' + 1)h + 1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height h. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth. [ABSTRACT FROM AUTHOR]

We apply the concept of formula treewidth and pathwidth to computation tree logic, linear temporal logic, and the full branching time logic. Several representations of formulas as graphlike structures are discussed, and corresponding notions of treewidth and pathwidth are introduced. As an application for such structures, we present a classification in terms of parametrised complexity of the satisfiability problem, where we make use of Courcelle's famous theorem for recognition of certain classes of structures. Our classification shows a dichotomy between W[1]-hard and fixed-parameter tractable operator fragments almost independently of the chosen graph representation. The only fragments that are proven to be fixed-parameter tractable (FPT) are those that are restricted to the X operator. By investigating Boolean operator fragments in the sense of Post's lattice, we achieve the same complexity as in the unrestricted case if the set of available Boolean functions can express the function "negation of the implication." Conversely, we show containment in FPT for almost all other clones. [ABSTRACT FROM AUTHOR]

Path decompositions of graphs are an important ingredient of dynamic programming algorithms for solving efficiently many NP-hard problems. Therefore, computing the pathwidth and associated path decomposition of graphs has both a theoretical and practical interest. In this article, we design a branch-and-bound algorithm that computes the exact pathwidth of graphs and a corresponding path decomposition. Our main contribution consists of several nontrivial techniques to reduce the size of the input graph (preprocessing) and to cut the exploration space during the search phase of the algorithm. We evaluate experimentally our algorithm by comparing it to existing algorithms of the literature. It appears from the simulations that our algorithm offers a significant gain with respect to previous work. In particular, it is able to compute the exact pathwidth of any graph with less than 60 nodes in a reasonable running time (≤ 10min on a standard laptop). Moreover, our algorithm achieves good performance when used as a heuristic (i.e., when returning best result found within bounded time limit). Our algorithm is not restricted to undirected graphs since it actually computes the directed pathwidth that generalizes the notion of pathwidth to digraphs. [ABSTRACT FROM AUTHOR]

Fast algorithms can be created for many graph problems when instances are confined to classes of graphs that are recursively constructed. This article first describes some basic conceptual notions regarding the design of such fast algorithms, and then the coverage proceeds through several recursive graph classes. Specific classes include trees, series-parallel graphs, k-terminal graphs, treewidth-k graphs, k-trees, partial k-trees, k-jackknife graphs, pathwidth-k graphs, bandwidth-k graphs, cutwidth-k graphs, branchwidth-k graphs, Halin graphs, cographs, cliquewidth-k graphs, k-NLC graphs, k-HB graphs, and rankwidth-k graphs. The definition of each class is provided. Typical algorithms are applied to solve problems on instances of most classes. Relationships between the classes are also discussed. [ABSTRACT FROM AUTHOR]