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The |$2$| -layer drawing model is a well-established paradigm to visualize bipartite graphs where vertices of the two parts lie on two horizontal lines and edges lie between these lines. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of |$k$| -planar graphs has been considered only for |$k=1$| in this context. We provide several contributions that address this gap in the literature. First, we show tight density bounds for the classes of |$2$| -layer |$k$| -planar graphs with |$k\in \{2,3,4,5\}$|⁠. Based on these results, we provide a Crossing Lemma for |$2$| -layer |$k$| -planar graphs, which then implies a general density bound for |$2$| -layer |$k$| -planar graphs. We prove this bound to be almost optimal with a corresponding lower bound construction. Finally, we study relationships between |$k$| -planarity and |$h$| -quasiplanarity in the |$2$| -layer model and show that |$2$| -layer |$k$| -planar graphs have pathwidth at most |$k+1$| while there are also |$2$| -layer |$k$| -planar graphs with pathwidth at least |$(k+3)/2$|⁠. [ABSTRACT FROM AUTHOR]

The circumference of a graph G with at least one cycle is the length of a longest cycle in G. A classic result of Birmel\'e [J. Graph Theory, 43 (2003), pp. 24--25] states that the treewidth of G is at most its circumference minus 1. In case G is 2-connected, this upper bound also holds for the pathwidth of G; in fact, even the treedepth of G is upper bounded by its circumference (Bria\'nski et al. [Treedepth vs circumference, Combinatorica, 43 (2023), pp. 659--664]). In this paper, we study whether similar bounds hold when replacing the circumference of G by its cocircumference, defined as the largest size of a bond in G, an inclusionwise minimal set of edges F such that G F has more components than G. In matroidal terms, the cocircumference of G is the circumference of the bond matroid of G. Our first result is the following "dual" version of Birmel'e's theorem: The treewidth of a graph G is at most its cocircumference. Our second and main result is an upper bound of 3k 2 on the pathwidth of a 2-connected graph G with cocircumference k. Contrary to circumference, no such bound holds for the treedepth of G. Our two upper bounds are best possible up to a constant factor. [ABSTRACT FROM AUTHOR]

Given two n-vertex graphs G1 and G2 of bounded treewidth, is there an n-vertex graph G of bounded treewidth having subgraphs isomorphic to G1 and G2? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if G1 is a binary tree and G2 is a ternary tree. We also provide an extensive study of cases where such "gluing"" is possible. In particular, we prove that if G1 has treewidth k and G2 has pathwidth \l, then there is an n-vertex graph of treewidth at most k + 3\l + 1 containing both G1 and G2 as subgraphs. [ABSTRACT FROM AUTHOR]

Given a graph G = (V , E) and an integer k, the Minimum Membership Dominating Set (MMDS) problem seeks to find a dominating set S ⊆ V of G such that for each v ∈ V , | N [ v ] ∩ S | is at most k. We investigate the parameterized complexity of the problem and obtain the following results for the MMDS problem. First, we show that the MMDS problem is NP-hard even on planar bipartite graphs. Next, we show that the MMDS problem is W[1]-hard for the parameter pathwidth (and thus, for treewidth) of the input graph. Then, for split graphs, we show that the MMDS problem is W[2]-hard for the parameter k. Further, we complement the pathwidth lower bound by an FPT algorithm when parameterized by the vertex cover number of input graph. In particular, we design a 2 O (v c) | V | O (1) time algorithm for the MMDS problem where vc is the vertex cover number of the input graph. Finally, we show that the running time lower bound based on ETH is tight for the vertex cover parameter. [ABSTRACT FROM AUTHOR]

We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph H to n-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new algorithms for counting and detecting graph patterns, and also for obtaining natural polynomial families which are complete for algebraic complexity classes VBP , V P , and VNP . We discover that, in the monotone setting, the formula complexity, the ABP complexity, and the circuit complexity of such polynomial families are exactly characterized by the treedepth, the pathwidth, and the treewidth of the pattern graph respectively. Furthermore, we establish a single, unified framework, using our characterization, to collect several known results that were obtained independently via different methods. For instance, we attain superpolynomial separations between circuits, ABPs, and formulas in the monotone setting, where the polynomial families separating the classes all correspond to well-studied combinatorial problems. Moreover, our proofs rediscover fine-grained separations between these models for constant-degree polynomials. [ABSTRACT FROM AUTHOR]

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]

The localization game is a pursuit-evasion game analogous to Cops and Robbers, where the robber is invisible and the cops send distance probes in an attempt to identify the location of the robber. We present a novel graph parameter called the localization capture time, which measures how long the localization game lasts assuming optimal play. We conjecture that the localization capture time is linear in the order of the graph, and show that the conjecture holds for graph families such as trees and interval graphs. We study bounds on the localization capture time for trees and its monotone property on induced subgraphs of trees and more general graphs. We give upper bounds for the localization capture time on the incidence graphs of projective planes. We finish with new bounds on the localization number and localization capture time using treewidth. [ABSTRACT FROM AUTHOR]

The Maximum Covering Location Problem (MCLP) is a well-studied problem in the field of operations research. Given a network with positive or negative demands on the nodes, a positive integer k, the MCLP seeks to find k potential facility centers in the network such that the neighborhood coverage is maximized. We study the variant of MCLP where edges of the network are subject to random failures due to some disruptive events. One of the popular models capturing the unreliable nature of the facility location is the linear reliability ordering (LRO) model. In this model, with every edge e of the network, we associate its survival probability 0 ≤ pe ≤ 1, or equivalently, its failure probability 1 − pe. The failure correlation in LRO is the following: If an edge e fails then every edge e ′ with p e ′ ≤ p e surely fails. The task is to identify the positions of k facilities that maximize the expected coverage. We refer to this problem as Expected Coverage problem. We study the Expected Coverage problem from the parameterized complexity perspective and obtain the following results. 1. For the parameter pathwidth, we show that the Expected Coverage problem is W[1]-hard. We find this result a bit surprising, because the variant of the problem with non-negative demands is fixed-parameter tractable (FPT) parameterized by the treewidth of the input graph. 2. We complement the lower bound by the proof that Expected Coverage is FPT being parameterized by the treewidth and the maximum vertex degree. We give an algorithm that solves the problem in time 2 O (tw log Δ) n O (1) , where tw is the treewidth, Δ is the maximum vertex degree, and n the number of vertices of the input graph. In particular, since Δ ≤ n, it means the problem is solvable in time n O (tw) , that is, is in XP parameterized by treewidth. [ABSTRACT FROM AUTHOR]

We study the problem of embedding shortest-path metrics of weighted graphs into ℓp spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of shortest path decomposition (SPD) depth is inductively defined: A (weighed) path graph has SPD depth 1. General graph has an SPD of depth k if it contains a shortest path whose deletion leads to a graph, each of whose components has SPD depth at most k - 1. In this paper we give an O(kmin{1/p,1/2})-distortion embedding for graphs of SPD depth at most k. This result is asymptotically tight for any fixed p > 1, while for p = 1 it is tight up to second order terms. As a corollary of this result, we show that graphs having pathwidth k embed into ℓp with distortion O(kmin{1/p,1/2}). For p = 1, this improves over the best previous bound of Lee and Sidiropoulos that was exponential in k; moreover, for other values of p it gives the first embeddings whose distortion is independent of the graph size n. Furthermore, we use the fact that planar graphs have SPD depth O(log n) to give a new proof that any planar graph embeds into ℓ1 with distortion O(√ log n). Our approach also gives new results for graphs with bounded treewidth, and for graphs excluding a fixed minor. [ABSTRACT FROM AUTHOR]

The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be NP -complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set. [ABSTRACT FROM AUTHOR]

In this paper we study the hardness of the k -Center problem on inputs that model transportation networks. For the problem, a graph G = (V , E) with edge lengths and an integer k are given and a center set C ⊆ V needs to be chosen such that | C | ≤ k . The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs. For each of these models, parameterized approximation algorithms have been shown to exist. We complement these results by proving that the k -Center problem is W[1]-hard on planar graphs of constant doubling dimension, where the parameter is the combination of the number of centers k, the highway dimension h, and the pathwidth p. Moreover, under the exponential time hypothesis there is no f (k , p , h) · n o (p + k + h) time algorithm for any computable function f. Thus it is unlikely that the optimum solution to k -Center can be found efficiently, even when assuming that the input graph abides to all of the above models for transportation networks at once! Additionally we give a simple parameterized (1 + ε) -approximation algorithm for inputs of doubling dimension d with runtime (k k / ε O (k d)) · n O (1) . This generalizes a previous result, which considered inputs in D-dimensional L q metrics. [ABSTRACT FROM AUTHOR]

The field of knowledge compilation establishes the tractability of many tasks by studying how to compile them to Boolean circuit classes obeying some requirements such as structuredness, decomposability, and determinism. However, in other settings such as intensional query evaluation on databases, we obtain Boolean circuits that satisfy some width bounds, e.g., they have bounded treewidth or pathwidth. In this work, we give a systematic picture of many circuit classes considered in knowledge compilation and show how they can be systematically connected to width measures, through upper and lower bounds. Our upper bounds show that bounded-treewidth circuits can be constructively converted to d-SDNNFs, in time linear in the circuit size and singly exponential in the treewidth; and that bounded-pathwidth circuits can similarly be converted to uOBDDs. We show matching lower bounds on the compilation of monotone DNF or CNF formulas to structured targets, assuming a constant bound on the arity (size of clauses) and degree (number of occurrences of each variable): any d-SDNNF (resp., SDNNF) for such a DNF (resp., CNF) must be of exponential size in its treewidth, and the same holds for uOBDDs (resp., n-OBDDs) when considering pathwidth. Unlike most previous work, our bounds apply to any formula of this class, not just a well-chosen family. Hence, we show that pathwidth and treewidth respectively characterize the efficiency of compiling monotone DNFs to uOBDDs and d-SDNNFs with compilation being singly exponential in the corresponding width parameter. We also show that our lower bounds on CNFs extend to unstructured compilation targets, with an exponential lower bound in the treewidth (resp., pathwidth) when compiling monotone CNFs of constant arity and degree to DNNFs (resp., nFBDDs). [ABSTRACT FROM AUTHOR]

We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalizes a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class. [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]

The crossing number is the smallest number of pairwise edge crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions, like treewidth or pathwidth. In this paper, we show that the crossing number is tractable (even in linear time) for maximal graphs of bounded pathwidth 3. The technique also shows that the crossing number and the rectilinear (a.k.a. straight-line) crossing number are identical for this graph class, and that we require only an O (n) × O (n) -grid to achieve such a drawing. Our techniques can further be extended to devise a 2-approximation for general graphs with pathwidth 3. One crucial ingredient here is that the crossing number of a graph with a separation pair can be lower-bounded using the crossing numbers of its cut-components, a result that may be interesting in its own right. Finally, we give a 4 w 3 -approximation of the crossing number for maximal graphs of pathwidth w . This is a constant approximation for bounded pathwidth. We complement this with an NP-hardness proof of the weighted crossing number already for pathwidth 3 graphs and bicliques K 3 , n . [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 work we consider two two-criteria optimization problems: given an input graph, the goal is to find its interval (or chordal) supergraph that minimizes the number of edges and its clique number simultaneously. For the interval supergraph, the problem can be restated as simultaneous minimization of the path width pw(G) and the profile p(G) of the input graph G. We prove that for an arbitrary graph G and an integer t ∈ {1, ... , pw(G) + 1}, there exists an interval supergraph G′ of G such that for its clique number it holds ω (G ′) ≤ (1 + 2 t) (pw (G) + 1) and the number of its edges is bounded by |E(G′)| ≤ (t + 2)p(G). In other words, the pathwidth and the profile of a graph can be simultaneously minimized within the factors of 1 + 2 t (plus a small constant) and t + 2, respectively. Note that for a fixed t, both upper bounds provide constant factor approximations. On the negative side, we show an example that proves that, for some graphs, there is no solution in which both parameters are optimal. In case of finding a chordal supergraph, the two corresponding graph parameters that reflect its clique size and number of edges are the treewidth and fill-in. We obtain that the treewidth and the fill-in problems are also 'orthogonal' in the sense that for some graphs, a solution that minimizes one of those parameters cannot minimize the other. As a motivating example, we recall graph searching games which illustrates a need of simultaneous minimization of these pairs of graph parameters. [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 give asymptotically exact values for the treewidth tw(G) of a random geometric graph G ∈ G(n, r) in [0, √n]². More precisely, let rc denote the threshold radius for the appearance of the giant component in G(n, r). We then show that for any constant 0 < r < rc, tw(G) = Θ(log n/log log n), and for c being sufficiently large, and r = r(n) ≥ c, tw(G) = Θ(r√n). Our proofs show that for the corresponding values of r the same asymptotic bounds also hold for the pathwidth and the treedepth of a random geometric graph. [ABSTRACT FROM AUTHOR]

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an n-vertex graph embedded on a surface of genus g with at most k crossings per edge has treewidth O(√(g + 1)(k + 1)n) and layered treewidth O((g + 1)k) and t hat these bounds are tight up to a constant factor. In the special case of g = 0, so-called k-planar graphs, the treewidth bound is O(√((k + 1)n), which is tight and improves upon a known O( (k + 1)3/4n1/2) bound. Analogous results are proved for map graphs defined with respect to any surface. Finally, we show that for g < m, every m-edge graph can be embedded on a surface of genus g with O((m/(g + 1)) log² g) crossings per edge, which is tight to a polylogarithmic factor. [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]

This paper tackles the well known graph searching problem, where a team of searchers aims at capturing an intruder in a network, modeled as a graph. This problem has been mainly studied for its relationship with the pathwidth of graphs. All variants of this problem assume that any node can be simultaneously occupied by several searchers. This assumption may be unrealistic, e.g., in the case of searchers modeling physical searchers, or may require each individual node to provide additional resources, e.g., in the case of searchers modeling software agents. We thus introduce and investigate exclusive graph searching, in which no two or more searchers can occupy the same node at the same time. As for the classical variants of graph searching, we study the minimum number of searchers required to capture the intruder. This number is called the exclusive search number of the considered graph. Exclusive graph searching appears to be considerably more complex than classical graph searching, for at least two reasons: (1) it does not satisfy the monotonicity property, and (2) it is not closed under minor. Moreover, we observe that the exclusive search number of a tree may differ exponentially from the values of classical search numbers (e.g., pathwidth). Nevertheless, we design a polynomial-time algorithm which, given any n-node tree T, computes the exclusive search number of T in time $$O(n^3)$$ . Moreover, for any integer k, we provide a characterization of the trees T with exclusive search number at most k. Finally, we prove that the ratio between the exclusive search number and the pathwidth of a graph is bounded by its maximum degree. [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]

In the mixed Chinese postman problem (MCPP), given a weighted mixed graph G (it may have both edges and arcs), our aim is to find a closed walk of minimum weight traversing each edge and arc at least once. The MCPP parameterized by the number of edges in G or the number of arcs in G is fixed-parameter tractable as proved by van Bevern et al. in 2014 and Gutin, Jones, and Sheng in 2014, respectively. Solving an open question of van Bevern et al., we show that somewhat unexpectedly the MCPP parameterized by the (undirected) treewidth of G is W[1]-hard. In fact, we prove that even the unweighted MCPP parameterized by the pathwidth of G is W[1]-hard. On the positive side, we show that MCPP parameterized by treedepth is fixed-parameter tractable (even with arbitrary integer weights). We are unaware of any widely studied graph parameters between pathwidth and treedepth and so our results provide a close characterization of the complexity of MCPP. [ABSTRACT FROM AUTHOR]

We give an algorithm for computing the directed pathwidth of a digraph with n vertices in $$O(1.89^{n})$$ time. This is the first algorithm with running time better than the straightforward $$O^{*}(2^n)$$ . As a special case, it computes the pathwidth of an undirected graph in the same amount of time, improving on the algorithm due to Suchan and Villanger which runs in $$O(1.9657^n)$$ time. [ABSTRACT FROM AUTHOR]

In the I ntervalizing Coloured Graphs problem, one must decide for a given graph G = ( V, E) with a proper vertex colouring of G whether G is the subgraph of a properly coloured interval graph. For the case that the number of colors is fixed, we give an exact algorithm that uses $2^{\mathcal {O}(n/\log n)}$ time. We also give an $\mathcal {O}^{\ast }(2^{n})$ algorithm for the case that the number of colors is not fixed. [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]

We prove a general reduction theorem which allows us to extend bounds for certain graph parameters on cubic graphs to bounds for general graphs taking into account the individual vertex degrees. As applications, we give an algorithm for Max $$2$$ -CSP whose complexity matches the algorithm of Scott and Sorkin in the case of $$d$$ -regular graphs, $$d \le 5$$ , but is otherwise faster. It also improves on the previously fastest known algorithm in terms of the average degree, given by Golovnev and Kutzkov. Also from the general theorem, we derive a bound for the pathwidth of a general graph which equals that of Fomin et al. and Gaspers for graphs of degree at most $$6$$ , but is smaller otherwise, and use this to give an improved exponential-space algorithm for Max $$2$$ -CSP. Finally we use the general result to give a faster algorithm for Max $$2$$ -CSP on claw-free graphs. [ABSTRACT FROM AUTHOR]

Birmele [ J Graph Theory 2003] proved that every graph with circumference t has treewidth at most [ABSTRACT FROM AUTHOR]

In this work we consider the edge searching problem for vertex-weighted graphs with arbitrarily fast and invisible fugitive. The weight function $${\omega }$$ provides for each vertex $$v$$ the minimum number of searchers required to guard $$v$$ , i.e., the fugitive may not pass through $$v$$ without being detected only if at least $${\omega }(v)$$ searchers are present at $$v$$ . This problem is a generalization of the classical edge searching problem, in which one has $${\omega }\equiv 1$$ . We assume that with a graph $$G$$ to be searched, there is associated a partition $$(V_1,\ldots ,V_t)$$ of its vertex set such that edges are allowed only within each $$V_i$$ and between two consecutive $$V_i$$ 's. We provide an algorithm for distributed monotone connected edge searching of such graphs, where the searchers are initially placed on an arbitrary vertex of $$G$$ and have no a priori knowledge on $$G$$ , but they have a sense of direction that lets them recognize whether an edge incident to already explored vertex in $$V_i$$ leads to a vertex in one of $$V_{i-1}, V_i$$ or $$V_{i+1}$$ . Starting from any vertex the algorithm uses at most $$3\cdot \max _{i=1,\ldots ,t}{\omega }(V_i)+1$$ searchers, where $${\omega }(V_i) = \sum _{v\in V_i}{\omega }(v)$$ . We also prove that this algorithm is best possible up to a small additive constant, that is, each distributed searching algorithm in worst case must use $$3\cdot \max _{i=1,\ldots ,t}{\omega }(V_i)-1$$ searchers for some graphs. [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]

We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth- $$2$$ outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on $$2n$$ vertices are bounded by $$O(n^{3})$$ and $$O(n^{10})$$ , in the convex and general case, respectively. We then apply similar methods to prove an $$n^{O(\log (n))}$$ upper bound on the Ramsey number of a path with $$n$$ ordered vertices. [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 will revisit the categorical notion of cospan decompositions of graphs and compare it to the well-known notions of path decomposition and tree decomposition from graph theory. More specifically, we will define several types of cospan decompositions with appropriate width measures and show that these width measures coincide with pathwidth and treewidth. Such graph decompositions of small width are used to efficiently decide graph properties, for instance via graph automata. Hence we will give an application by defining graph-accepting tree automata, thus integrating previous work by Courcelle into the setting of cospan decompositions. Furthermore we will show that regardless of whether we consider path or tree decompositions, we arrive at the same notion of recognizability. [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]

It is proven that the connected pathwidth of any graph G is at most 2 · pw(G)+1, where pw(G) is the pathwidth of G. The method is constructive, i.e., it yields an efficient algorithm that for a given path decomposition of width k computes a connected path decomposition of width at most 2k+1. The running time of the algorithm is O(dk²), where d is the number of "bags" in the input path decomposition. The motivation for studying connected path decompositions comes from the connection between the pathwidth and the search number of a graph. One of the advantages of the above bound for connected pathwidth is an inequality cs(G) ≤ 2s(G) + 3, where cs(G) and s(G) are the connected search number and the search number of G, respectively. Moreover, the algorithm presented in this work can be used to convert a given search strategy using k searchers into a (monotone) connected one using 2k + 3 searchers and starting at an arbitrary homebase. [ABSTRACT FROM AUTHOR]

In this paper, we study the step-wise tile assembly model introduced by Reif (in: Based computers III, vol 48 of DIMACS, ) in which shape is assembled in multiple steps. In each step the partially built structure is exposed to a new set of tiles. We show that an N × N square can be assembled in N steps using a constant number of tile types. In general, we show that a constant number of tile types (24) is sufficient to assemble a large class of shapes in n steps, where n is the number of tiles of the shape. This class includes all shapes obtained from any shape by scaling by a factor of 2, in which case only 14 tile types suffices. For general shapes, we show that the tile complexity of this model is related to the monotone connected node search number of a spanning tree of the shape assuming the number of steps is n. [ABSTRACT FROM AUTHOR]

The Pathwidth One Vertex Deletion (POVD) problem asks whether, given an undirected graph G and an integer k, one can delete at most k vertices from G so that the remaining graph has pathwidth at most 1. The question can be considered as a natural variation of the extensively studied Feedback Vertex Set (FVS) problem, where the deletion of at most k vertices has to result in the remaining graph having treewidth at most 1 (i.e., being a forest). Recently Philip et al. (WG, Lecture Notes in Computer Science, vol. 6410, pp. 196-207, ) initiated the study of the parameterized complexity of POVD, showing a quartic kernel and an algorithm which runs in time 7 n. In this article we improve these results by showing a quadratic kernel and an algorithm with time complexity 4.65 n, thus obtaining almost tight kernelization bounds when compared to the general result of Dell and van Melkebeek (STOC, pp. 251-260, ACM, New York, ). Techniques used in the kernelization are based on the quadratic kernel for FVS, due to Thomassé (ACM Trans. Algorithms 6(2), ). [ABSTRACT FROM AUTHOR]

We present a distributed algorithm to compute the node search number in trees. This algorithm extends the centralized algorithm proposed by Ellis et al. (Inf. Comput. 113(1):50-79, ). It can be executed in an asynchronous environment, requires an overall computation time of O( nlog n), and n messages of log n+4 bits each. The main contribution of this work lies in the data structure proposed to design our algorithm, called hierarchical decomposition. This simple and flexible data structure is used for four operations: updating the node search number after addition or deletion of any tree-edges in a distributed fashion; computing it in a tree whose edges are added sequentially and in any order; computing other graph invariants such as the process number and the edge search number, by changing only initialization rules; extending our algorithms for trees and forests of unknown size (using messages of up to 2log n+5 bits). [ABSTRACT FROM AUTHOR]

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]