Search

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 study the combinatorial FIFO Stack-Up problem. In the delivery industry, bins have to be moved from conveyor belts onto pallets. Given is a list Q of k sequences of labeled bins and a positive integer p. The goal is to palletize the bins by iteratively removing the first bin of one of the k sequences and putting it onto a pallet located at one of p stack-up places. Each of these pallets has to contain bins of only one label, bins of different labels have to be placed on different pallets. After all bins of one label have been removed from the given sequences, the corresponding stack-up place becomes available for a pallet of bins of another label. The FIFO Stack-Up problem asks whether such a processing is possible. The FIFO Stack-Up problem is known to be NP-complete even if the sequences of Q contain together at most c Q = 6 bins per pallet. This implies that the problem is also hard if all bins of every pallet are distributed to at most d Q = 6 sequences. In this paper we strengthen the hardness to the cases c Q = 5 and d Q = 3. We also show the hardness for the practical case where the number of sequences is smaller than the number of pallets, and we point out differences to related problems like 3D Bin Packing and Train Marshalling. [ABSTRACT FROM AUTHOR]

A connected path decomposition of a simple graph G is a path decomposition (X 1 , ... , X l) such that the subgraph of G induced by X 1 ∪ ⋯ ∪ X i is connected for each i ∈ { 1 , ... , l }. The connected pathwidth of G is then the minimum width over all connected path decompositions of G. We prove that for each fixed k , the connected pathwidth of any input graph can be computed in polynomial-time. This answers an open question raised by Fedor V. Fomin during the GRASTA 2017 workshop, since connected pathwidth is equivalent to the connected (monotone) node search game. [ABSTRACT FROM AUTHOR]

The topological theory for semi-complete digraphs, pioneered by Chudnovsky, Fradkin, Kim, Scott, and Seymour [10–12,28,43,39] , concentrates on the interplay between the most important width measures — cutwidth and pathwidth — and containment relations like topological/minor containment or immersion. We propose a new approach to this theory that is based on outdegree orderings and new families of obstacles for cutwidth and pathwidth. Using the new approach we are able to reprove the most important known results in a unified and simplified manner, as well as provide multiple improvements. Notably, we obtain a number of efficient approximation and fixed-parameter tractable algorithms for computing width measures of semi-complete digraphs, as well as fast fixed-parameter tractable algorithms for testing containment relations in the semi-complete setting. As a direct corollary of our work, we also derive explicit and essentially tight bounds on duality relations between width parameters and containment orderings in semi-complete digraphs. [ABSTRACT FROM AUTHOR]

In this paper, we provide results for the search number of the Cartesian product of graphs. We consider graphs on opposing ends of the spectrum: paths and cliques. Our main result determines the pathwidth of the product of cliques and provides a lower bound for the search number of the product of cliques. A consequence of this result is a bound for the search number of the product of arbitrary graphs G and H based on their respective clique numbers. [ABSTRACT FROM AUTHOR]

The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs . We show that determining the treewidth of the line graph of a graph G is equivalent to determining the minimum vertex congestion of an embedding of G into a tree. Using this result, we prove sharp lower bounds in terms of both the minimum degree and average degree of G . These results are precise enough to exactly determine the treewidth of the line graph of a complete graph and other interesting examples. We also improve the best known upper bound on the treewidth of a line graph. Analogous results are proved for pathwidth. [ABSTRACT FROM AUTHOR]

So-called ordered variants of the classical notions of pathwidth and treewidth are introduced and proposed as proof theoretically meaningful complexity measures for the directed acyclic graphs underlying proofs. Ordered pathwidth is roughly the same as proof space and the ordered treewidth of a proof is meant to serve as a measure of how far it is from being treelike. Length-space lower bounds for k -DNF refutations are generalized to arbitrary infinity axioms and strengthened in that the space measure is relaxed to ordered treewidth. [ABSTRACT FROM AUTHOR]

We consider the complexity of the firefighter problem where a budget of b ≥ 1 firefighters are available at each time step. This problem is known to be NP-complete even on trees of degree at most three and b = 1 [14] and on trees of bounded degree ( b + 3 ) for any fixed b ≥ 2 [4] . In this paper we provide further insight into the complexity landscape of the problem by showing a complexity dichotomy result with respect to the parameters pathwidth and maximum degree of the input graph. More precisely, first, we prove that the problem is NP-complete even on trees of pathwidth at most three for any b ≥ 1 . Then we show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter “pathwidth” and “maximum degree” of the input graph. Finally, we show that the problem remains NP-complete on very dense graphs, namely co-bipartite graphs, but is fixed-parameter tractable with respect to the parameter “cluster vertex deletion”. [ABSTRACT FROM AUTHOR]

In Graph Searching, a team of searchers aims at capturing an invisible fugitive moving arbitrarily fast in a graph. Equivalently, the searchers try to clear a contaminated network. The problem is to compute the minimum number of searchers required to accomplish this task. Several variants of Graph Searching have been studied mainly because of their close relationship with the pathwidth of a graph. In this paper, we study the complexity of the Exclusive Graph Searching variant. We show that the problem is NP-hard in planar graphs and it can be solved in linear-time in the class of cographs. We also show that monotone Exclusive Graph Searching is NP-complete in split graphs where Pathwidth is known to be solvable in polynomial time. Moreover, we prove that monotone Exclusive Graph Searching is in P in a subclass of star-like graphs where Pathwidth is known to be NP-hard. [ABSTRACT FROM AUTHOR]

After the number of vertices, Vertex Cover Number is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex Cover Number . Here we consider the treewidth and pathwidth problems parameterized by k , the size of a minimum vertex cover of the input graph. We show that the pathwidth and treewidth can be computed in O ∗ ( 3 k ) time. This complements recent polynomial kernel results for treewidth and pathwidth parameterized by the Vertex Cover Number . [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]

Non-deterministic graph searching was introduced by Fomin et al. to provide a unified approach for pathwidth, treewidth, and their interpretations in terms of graph searching games. Given q ≥ 0 , the q -limited search number, s q ( G ) , of a graph G is the smallest number of searchers required to capture an invisible fugitive in G , when the searchers are allowed to know the position of the fugitive at most q times. The search parameter s 0 ( G ) corresponds to the pathwidth of a graph G , and s ∞ ( G ) to its treewidth. Determining s q ( G ) is NP-complete for any fixed q ≥ 0 in general graphs and s 0 ( T ) can be computed in linear time in trees, however the complexity of the problem on trees has been unknown for any q > 0 . We introduce a new variant of graph searching called restricted non-deterministic. The corresponding parameter is denoted by rs q and is shown to be equal to the non-deterministic graph searching parameter s q for q = 0 , 1 , and at most twice s q for any q ≥ 2 (for any graph G ). Our main result is a polynomial time algorithm that computes rs q ( T ) for any tree T and any q ≥ 0 . This provides a 2-approximation of s q ( T ) for any tree T , and shows that the decision problem associated to s 1 is polynomial in the class of trees. Our proofs are based on a new decomposition technique for trees which might be of independent interest. [ABSTRACT FROM AUTHOR]

Given a connected outerplanar graph G of pathwidth p , we give an algorithm to add edges to G to get a supergraph of G , which is 2-vertex-connected, outerplanar and of pathwidth O ( p ) . This settles an open problem raised by Biedl [1] , in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two-dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl [1] for 2-vertex-connected outerplanar graphs will work for all outer planar graphs. [ABSTRACT FROM AUTHOR]

Research on the similarity of a graph to being a tree—called the treewidth of the graph—has seen an enormous rise within the last decade, but a practically fast algorithm for this task has been discovered only recently by Tamaki (ESA 2017). It is based on dynamic programming and makes use of the fact that the number of positive subinstances is typically substantially smaller than the number of all subinstances. Algorithms producing only such subinstances are called positive-instance driven (PID). The parameter treedepth has a similar story. It was popularized through the graph sparsity project and is theoretically well understood—but the first practical algorithm was discovered only recently by Trimble (IPEC 2020) and is based on the same paradigm. We give an alternative and unifying view on such algorithms from the perspective of the corresponding configuration graphs in certain two-player games. This results in a single algorithm that can compute a wide range of important graph parameters such as treewidth, pathwidth, and treedepth. We complement this algorithm with a novel randomized data structure that accelerates the enumeration of subproblems in positive-instance driven algorithms. [ABSTRACT FROM AUTHOR]

Abstract: For each non-negative integer , we provide all outerplanar obstructions for the class of graphs whose cycle matroid has pathwidth at most . Our proof combines a decomposition lemma for proving lower bounds on matroid pathwidth and a relation between matroid pathwidth and linear width. Our results imply the existence of a linear algorithm that, given an outerplanar graph, outputs its matroid pathwidth. [Copyright &y& Elsevier]

Abstract: We prove that if a tournament has pathwidth then it has θ vertices that are pairwise θ-connected. As a consequence of this and previous results, we obtain that for every set of tournaments the following are equivalent: [•] there exists k such that every member of has pathwidth at most k, [•] there is a digraph H such that no subdivision of H is a subdigraph of any member of , [•] there exists k such that for each , there do not exist k vertices of T that are pairwise k-connected. As a further consequence, we obtain a polynomial-time algorithm to test whether a tournament contains a subdivision of a fixed digraph H as a subdigraph. [Copyright &y& Elsevier]

Abstract: The problems of (classical) searching and connected searching of weighted trees are known to be computationally hard. In this work we give a polynomial-time 3-approximation algorithm that finds a connected search strategy of a given weighted tree. This in particular yields constant factor approximation algorithms for the (non-connected) classical searching problems and for the weighted pathwidth problem for this class of graphs. [Copyright &y& Elsevier]

Abstract: An acyclic coloring of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of star coloring requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding an optimal acyclic and star coloring of a cograph. If the graph is given in the form of a cotree, the algorithm runs in time. We also show that the acyclic chromatic number, the star chromatic number, the treewidth plus 1, and the pathwidth plus 1 are all equal for cographs. [Copyright &y& Elsevier]

Abstract: We introduce the process number of a digraph as a tool to study rerouting issues in wdm networks. This parameter is closely related to the vertex separation (or pathwidth). We consider the recognition and the characterization of (di)graphs with small process numbers. In particular, we give a linear time algorithm to recognize (and process) graphs with process number at most 2, along with a characterization in terms of forbidden minors, and a structural description. As for digraphs with process number 2, we exhibit a characterization that allows one to recognize (and process) them in polynomial time. [Copyright &y& Elsevier]

Abstract: We study the bandwidth and the pathwidth of multi-dimensional grids. It can be shown for grids, that these two parameters are equal to a more basic graph parameter, the vertex boundary width. Using this fact, we determine the bandwidth and the pathwidth of three-dimensional grids, which were known only for the cubic case. As a by-product, we also determine the two parameters of multi-dimensional grids with relatively large maximum factors. [Copyright &y& Elsevier]

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 traditional edge searching one tries to clean all of the edges in a graph employing the least number of searchers. It is assumed that each edge of the graph initially has a weight equal to one. In this paper we modify the problem and introduce the Weighted Edge Searching Problem by considering graphs with arbitrary positive integer weights assigned to its edges. We give bounds on the weighted search number in terms of related graph parameters including pathwidth. We characterize the graphs for which two searchers are sufficient to clear all edges. We show that for every weighted graph the minimum number of searchers needed for a not-necessarily-monotonic weighted edge search strategy is enough for a monotonic weighted edge search strategy, where each edge is cleaned only once. This result proves the NP-completeness of the problem. [Copyright &y& Elsevier]

Abstract: Let be a finite, simple and undirected graph. For , let be the edge boundary of . Given an integer , , let the edge isoperimetric value of at be defined as . The edge isoperimetric peak of is defined as . Let denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete -ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete -ary trees, Discrete Mathematics, in press (doi:10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees. The depth of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth (denoted as ), and where , are constants. For a complete -ary tree of depth (denoted as ) and where is a constant, we show that and where , are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete -ary tree. Let be a finite, connected and rooted tree — the root being the vertex . Define a weight function where the weight of a vertex is the number of its successors (including itself) and let the weight index be defined as the number of distinct weights in the tree, i.e . For a positive integer , let . We show that . [Copyright &y& Elsevier]

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]

Abstract: Search games are attractive for their correspondence with classical width parameters. For instance, the invisible search number (a.k.a. node search number) of a graph is equal to its pathwidth plus 1, and the visible search number of a graph is equal to its treewidth plus 1. The connected variants of these games ask for search strategies that are connected, i.e., at every step of the strategy, the searched part of the graph induces a connected subgraph. We focus on monotone search strategies, i.e., strategies for which every node is searched exactly once. The monotone connected visible search number of an n-node graph is at most times its visible search number. First, we prove that this logarithmic bound is tight. Precisely, we prove that there is an infinite family of graphs for which the ratio monotone connected visible search number over visible search number is . Second, we prove that, as opposed to the non-connected variant of visible graph searching, “recontamination helps” for connected visible search. Precisely, we prove that, for any , there exists a graph with connected visible search number at most k, and monotone connected visible search number >k [Copyright &y& Elsevier]

Abstract: We present proofs of lower bounds on the node search number of some grid-like graphs including two-dimensional grids, cylinders, tori and a variation we call “orb-webs”. Node search number is equivalent to pathwidth and vertex separation, which are all important graph parameters. Since matching upper bounds are not difficult to obtain, this implies that the pathwidth of these graphs is easily computed, because the bounds are simple functions of the graph dimensions. We also show matching upper and lower bounds on the node search number of equidimensional tori which are one less than the obvious upper bound. [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]

Abstract: For a given graph , the interval completion problem of G is to find an edge set F such that the supergraph of G is an interval graph and is minimum. It has been shown that it is equivalent to the minimum sum cut problem, the profile minimization problem and a kind of graph searching problems. Furthermore, it has applications in computational biology, archaeology, and clone fingerprinting. In this paper, we show that it is NP-complete on split graphs and propose an efficient algorithm on primitive starlike graphs. [Copyright &y& Elsevier]

Abstract: The d-dimensional hypercube, , is the graph on vertices, which correspond to the d-vectors whose components are either 0 or 1, two of the vertices being adjacent when they differ in just one coordinate. The notion of Hamming graphs (denoted by ) generalizes the notion of hypercubes: The vertices correspond to the d-vectors where the components are from the set , and two of the vertices are adjacent if and only if the corresponding vectors differ in exactly one component. In this paper we show that the and the . [Copyright &y& Elsevier]

Abstract: We present embedding algorithms (subgraph isomorphism and its generalizations) for classes of graphs of bounded pathwidth, where n is the number of vertices in the graph. These include the first polynomial-time algorithm for minor containment and the first algorithm (c a constant independent of k) for topological embedding of graphs from subclasses of partial k-trees, as well as an algorithm for subgraph isomorphism. Of independent interest are structural properties of k-connected graphs of bounded pathwidth on which our algorithms are based. We also describe special cases which reduce to various generalizations of string matching, permitting more efficient solutions. Finally, we describe algorithms for solving these problems on arbitrary graphs of pathwidth at most k. [Copyright &y& Elsevier]

Abstract: Motivated by the desire to speed up dynamic programming algorithms for graphs of bounded treewidth, we initiate a study of the tradeoff between width and pathwidth of tree-decompositions. We therefore investigate the catwidth parameter which is the minimum width of any tree-decomposition of a graph G when the pathwidth of the tree T is 1. The catwidth parameter lies between the treewidth and the pathwidth of the graph, , and just as treewidth relates to chordal graphs and pathwidth relates to interval graphs, catwidth relates to what we call catval graphs. We introduce the notion of an extended asteroidal triple (XAT) and characterize catval graphs as the XAT-free chordal graphs. We provide alternative characterizations of these graphs, show that there are graph classes for which the various parameters differ by an arbitrary amount, and consider algorithms for computing catwidth. [Copyright &y& Elsevier]

We describe an O(nlogn) algorithm for the computation of the vertex separation of unicyclic graphs. The algorithm also computes a linear layout with optimal vertex separation in the same time bound. Pathwidth, node search number and vertex separation are different ways of defining the same notion. Path decompositions and search strategies can be derived from linear layouts. The algorithm applies existing, linear time, techniques for the computation of the vertex separation of trees together with corresponding optimal layouts. We reformulate the earlier work on the linear time computation of optimal layouts. A polynomial time algorithm for the problem on unicyclic graphs can be inferred from existing more general methods for graphs of fixed treewidth, since unicyclic graphs have treewidth two, but the time complexity of the resulting method would seem to be an inordinately high order polynomial. Our algorithm we claim is “practical.” The addition of one edge to a tree does seem to require a considerably more elaborate algorithm. [Copyright &y& Elsevier]

Let P(G; x; y) be the number of vertex colorings &phi : V → {1,...,x} of an undirected graph G = (V, E) such that for all edges { u, v} ≠ E the relations &phi(v) ≤ y and φ(v) ≤ y implyφ(u) ≠ φ(v). We show that P(G; x, y) is a polynomial in x and y which is closely related to Stanley's chromatic symmetric function, and which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of G. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex, and one in terms of the lattice of forbidden colorings. Finally, we give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that P(G; x, y) can be evaluated in polynomial time for trees and graphs of restricted pathwidth. [ABSTRACT FROM AUTHOR]

The interval degree id(G) of a graph G is defined to be the smallest max-degree of any interval supergraphs of G. One of the reasons for our interest in this parameter is that the bandwidth of a graph is always between id(G)/2 and id(G). We prove also that for any graph G the interval degree of G is at least the pathwidth of G2. We show that if G is an AT-free claw-free graph, then the interval degree of G is equal to the clique number of G2 minus one. Finally, we show that there is a polynomial time algorithm for computing the interval degree of AT-free claw-free graphs. [Copyright &y& Elsevier]

Let P(G;x,y) be the number of vertex colorings Φ : V → {1,...,x} of an undirected graph G = (V,E) such that for all edges {u,v} ∈ E the relations Φ(u) ≤ y and Φ(v) ≤ y imply Φ(u) ≠ Φ(v). We show that P(G;x,y) is a polynomial in x and y which is closely related to Stanley's chromatic symmetric function, and which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of G. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex, and one in terms of the lattice of forbidden colorings. Finally, we give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that P(G;x, y) can be evaluated in polynomial time for trees and graphs of restricted pathwidth. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that the maximum pagenumber of the graphs with pathwidth k is k and that the maximum pagenumber of the graphs with strong pathwidth k is in between ⌈3(k−1)/2⌉ and 3⌈k/2⌉. [Copyright &y& Elsevier]

It is known that the class of graphs with treewidth (resp. pathwidth) bounded by a constant w can be characterized by a finite obstruction set obs(TW(w)) (resp. obs(PW(w))). These obstruction sets are known for w⩽3 so far. In this paper we give a structural characterization of graphs from obs(TW(w)) (resp. obs(PW(w))) with a fixed number of vertices in terms of subgraphs of the complement. Our approach also essentially simplifies known characterization of graphs from obs(TW(w)) (resp. obs(PW(w))) with (w+3) vertices.Also for any w⩾3 a graph from obs(TW(w))&z.drule;obs(PW(w)) is constructed, that solves an open problem. [Copyright &y& Elsevier]

As technology advances and the spreading of wireless devices grows, the establishment of interconnection networks is becoming crucial. Main activities that involve most of the people concern retrieving and sharing information from everywhere. In heterogeneous networks, devices can communicate by means of multiple interfaces. The choice of the most suitable interfaces to activate (switch-on) at each device results in the establishment of different connections. A connection is established when at its endpoints the devices activate at least one common interface. Each interface is assumed to consume a specific percentage of energy for its activation. This is referred to as the cost of an interface. Due to energy consumption issues, and the fact that most of the devices are battery powered, special effort must be devoted to suitable solutions that prolong the network lifetime. In this paper, we consider the so-called p-Coverage problem where each device can activate at most p of its available interfaces in order to establish all the desired connections of a given network of devices. As the problem has been shown to be NP -hard even for p = 2 and unitary costs of the interfaces, algorithmic design activities have focused in particular topologies where the problem is optimally solvable. Following this trend, we first show that the problem is polynomially solvable for graphs (modeling the underlying network) of bounded treewidth by means of the Courcelle's theorem. Then, we provide two optimal polynomial time algorithms to solve the problem in two subclasses of graphs with bounded treewidth that are graphs of bounded pathwidth and graphs of bounded carvingwidth. The two solutions are obtained by means of dynamic programming techniques. [ABSTRACT FROM AUTHOR]