IA Scholar Query: Hitting forbidden subgraphs in graphs of bounded treewidth.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 08 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Hitting forbidden induced subgraphs on bounded treewidth graphs
https://scholar.archive.org/work/6iykiie74bdhdjptc5ydxh6xqy
For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ⊆ V(G) such that G∖ S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion can be solved in time f_H(t) · n^O(1) assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively. We show that f_H(t) = 2^O(t^h-2) for every graph H on h ≥ 3 vertices, and that f_H(t) = 2^O(t) if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function f_H(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then f_H(t) = 2^Θ(t^h-2). We also show that f_H(t) = 2^Ω(t^h) when H=K_h,h, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function f_C_4(t) for the subgraph version. Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V(H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function f_H(t) for every connected graph H on h vertices: if h≤ 2 the problem can be solved in polynomial time; if h≥ 3, f_H(t) = 2^Θ(t) if H is a clique, and f_H(t) = 2^Θ(t^h-2) otherwise.Ignasi Sau, Uéverton S. Souzawork_6iykiie74bdhdjptc5ydxh6xqyThu, 08 Sep 2022 00:00:00 GMTDeterministic Fault-Tolerant Connectivity Labeling Scheme with Adaptive Query Processing Time
https://scholar.archive.org/work/l6lroggr7zgwflegmqskk5itue
The f-fault-toleratant connectivity labeling (f-FTC labeling) is a scheme of assigning each vertex and edge with a small-size label such that one can determine the connectivity of two vertices s and t under the presence of at most f faulty edges only from the labels of s, t, and the faulty edges. This paper presents a new deterministic f-FTC labeling scheme attaining O(f^2 polylog(n))-bit label size and a polynomial construction time, which settles the open problem left by Dory and Parter [PODC'21]. The key ingredient of our construction is to develop a deterministic counterpart of the graph sketch technique by Ahn, Guha, and McGreger [SODA'12], via some natural connection with the theory of error-correcting codes. This technique removes one major obstacle in de-randomizing the Dory-Parter scheme. The whole scheme is obtained by combining this technique with a new deterministic graph sparsification algorithm derived from the seminal ϵ-net theory, which is also of independent interest. The authors believe that our new technique is potentially useful in the future exploration of more efficient FTC labeling schemes and other related applications based on graph sketches. An interesting byproduct of our result is that one can obtain an improved randomized f-FTC labeling scheme attaining adaptive query processing time (i.e., the processing time does not depend on f, but only on the actual number of faulty edges). This scheme is easily obtained from our deterministic scheme just by replacing the graph sparsification part with the conventional random edge sampling.Taisuke Izumi, Yuval Emek, Tadashi Wadayama, Toshimitsu Masuzawawork_l6lroggr7zgwflegmqskk5itueWed, 24 Aug 2022 00:00:00 GMTOffensive Alliances in Graphs
https://scholar.archive.org/work/ck5reoup55dk5febs2dcqjk4ta
A set S⊆ V of vertices is an offensive alliance in an undirected graph G=(V,E) if each v∈ N(S) has at least as many neighbours in S as it has neighbours (including itself) not in S. We study the classical and parameterized complexity of the Offensive Alliance problem, where the aim is to find a minimum size offensive alliance. Our focus here lies on natural parameter as well as parameters that measure the structural properties of the input instance. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, treewidth, pathwidth, and treedepth of the input graph; we thereby resolve an open question stated by Bernhard Bliem and Stefan Woltran (2018) concerning the complexity of Offensive Alliance parameterized by treewidth, (2) unless ETH fails, Offensive Alliance problem cannot be solved in time 𝒪^*(2^o(k log k)) where k is the solution size, (3) Offensive Alliance problem does not admit a polynomial kernel parameterized by solution size and vertex cover of the input graph. On the positive side we prove that (4) Offensive Alliance can be solved in time 𝒪^*(vc(G)^𝒪(vc(G))) where vc(G) is the vertex cover number of the input graph. In terms of classical complexity, we prove that (5) Offensive Alliance problem cannot be solved in time 2^o(n) even when restricted to bipartite graphs, unless ETH fails, (6) Offensive Alliance problem cannot be solved in time 2^o(√(n)) even when restricted to apex graphs, unless ETH fails. We also prove that (7) Offensive Alliance problem is NP-complete even when restricted to bipartite, chordal, split and circle graphs.Ajinkya Gaikwad, Soumen Maitywork_ck5reoup55dk5febs2dcqjk4taFri, 05 Aug 2022 00:00:00 GMTOn Upward-Planar L-Drawings of Graphs
https://scholar.archive.org/work/d7p6jsgyjrc3xjb5bpxlqgykei
In an upward-planar L-drawing of a directed acyclic graph (DAG) each edge e is represented as a polyline composed of a vertical segment with its lowest endpoint at the tail of e and of a horizontal segment ending at the head of e. Distinct edges may overlap, but not cross. Recently, upward-planar L-drawings have been studied for st-graphs, i.e., planar DAGs with a single source s and a single sink t containing an edge directed from s to t. It is known that a plane st-graph, i.e., an embedded st-graph in which the edge (s,t) is incident to the outer face, admits an upward-planar L-drawing if and only if it admits a bitonic st-ordering, which can be tested in linear time. We study upward-planar L-drawings of DAGs that are not necessarily st-graphs. On the combinatorial side, we show that a plane DAG admits an upward-planar L-drawing if and only if it is a subgraph of a plane st-graph admitting a bitonic st-ordering. This allows us to show that not every tree with a fixed bimodal embedding admits an upward-planar L-drawing. Moreover, we prove that any acyclic cactus with a single source (or a single sink) admits an upward-planar L-drawing, which respects a given outerplanar embedding if there are no transitive edges. On the algorithmic side, we consider DAGs with a single source (or a single sink). We give linear-time testing algorithms for these DAGs in two cases: (i) when the drawing must respect a prescribed embedding and (ii) when no restriction is given on the embedding, but it is biconnected and series-parallel.Patrizio Angelini, Steven Chaplick, Sabine Cornelsen, Giordano Da Lozzowork_d7p6jsgyjrc3xjb5bpxlqgykeiWed, 03 Aug 2022 00:00:00 GMTVertex Deletion Parameterized by Elimination Distance and Even Less
https://scholar.archive.org/work/sdzr3cd7lrdmnjd5v32mha6lde
We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems either focus on the parameterization by solution size, detecting solutions of size k in time f(k) · n^O(1), or width parameterizations, finding arbitrarily large optimal solutions in time f(w) · n^O(1) for some width measure w like treewidth. We unify these lines of research by presenting FPT algorithms for parameterizations that can simultaneously be arbitrarily much smaller than the solution size and the treewidth. We consider two classes of parameterizations which are relaxations of either treedepth of treewidth. They are related to graph decompositions in which subgraphs that belong to a target class H (e.g., bipartite or planar) are considered simple. First, we present a framework for computing approximately optimal decompositions for miscellaneous classes H. Namely, if the cost of an optimal decomposition is k, we show how to find a decomposition of cost k^O(1) in time f(k) · n^O(1). This is applicable to any graph class H for which the corresponding vertex-deletion problem admits a constant-factor approximation algorithm or an FPT algorithm paramaterized by the solution size. Secondly, we exploit the constructed decompositions for solving vertex-deletion problems by extending ideas from algorithms using iterative compression and the finite state property. For the three mentioned vertex-deletion problems, and all problems which can be formulated as hitting a finite set of connected forbidden (a) minors or (b) (induced) subgraphs, we obtain FPT algorithms with respect to both studied parameterizations.Bart M. P. Jansen, Jari J. H. de Kroon, Michał Włodarczykwork_sdzr3cd7lrdmnjd5v32mha6ldeMon, 18 Jul 2022 00:00:00 GMTFixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation
https://scholar.archive.org/work/2clzgjqoljfplksmheh4qpqzga
We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem, given a directed graph G, pairs of vertices (called terminals) (s_1,t_1), (s_2,t_2), and (s_3,t_3), and an integer k, asks to find a set of at most k non-terminal vertices in G that intersect all s_1t_1-paths, all s_2t_2-paths, and all s_3t_3-paths. The parameterized complexity of this case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the 4-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain.We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendes, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the "middle" box in the grid minor can be proclaimed irrelevant – not contained in the sought solution – and thus reduced.Meike Hatzel and Lars Jaffke and Paloma T. Lima and Tomáš Masařík and Marcin Pilipczuk and Roohani Sharma and Manuel Sorgework_2clzgjqoljfplksmheh4qpqzgaFri, 15 Jul 2022 00:00:00 GMTTemporal graph exploration: restrictions and relaxations
https://scholar.archive.org/work/kxeukfp2efctjgg3en7gcwhpk4
This thesis considers the problem of exploring temporal graphs. A temporal graph G = hG1; :::;GLi of order n is a sequence of L undirected graphs (or layers) indexed by the timesteps t 2 f1; : : : ;Lg, such that V (G1) = V (G) and E(Gt) E(G) for some underlying graph G with order n. To explore G is to visit each vertex at least once via a sequence of edge-traversals (called an exploration schedule), with each consecutive edge traversed during a timestep strictly greater than the last. The arrival time of an the timestep during which the last unvisited vertex is reached for the first time.There exists an algorithm producing exploration schedules with arrival time O(n2) for any always-connected (i.e., Gt is connected for all t 2 f1; : : : ;Lg) temporal graph, and an infinite family F of always-connected temporal graphs for which any exploration schedule has arrival time (n2) [38, 86]. We isolate a number of characteristics held by the members of F and prove lower/upper bounds on the arrival time of exploration schedules for temporal graphs that are restricted from possessing them. First, we consider structural restrictions in which an input temporal graph has (1) degree upper bounded by in each layer; and (2) at most k edges 'missing' from the underlying graph in each layer; subquadratic upper bounds are proved in each case. We then consider 'relaxed' exploration schedules that can traverse a ?nite number of edges ( 1) in each timestep, focusing on the cases when 2 or n=k traversals are allowed. We also consider, from a complexity standpoint, a number of relaxed problem variants, in which (1) less than n vertices are required to be explored by a candidate, and (2) an unlimited but ?nite number of edge traversals can be made by a candidate exploration schedule, providing both FPT-membership results and hardness/NP-completeness results.Jakob T. Spoonerwork_kxeukfp2efctjgg3en7gcwhpk4Thu, 14 Jul 2022 00:00:00 GMTGraph Product Structure for h-Framed Graphs
https://scholar.archive.org/work/2nvonsl6svaidjkrnjopigafky
Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of a bounded treewidth graph, and allows to lift combinatorial results for bounded treewidth graphs to graph classes for which the product structure holds, such as to planar graphs [Dujmović et al., J. ACM, 67(4), 22:1-38, 2020]. In this paper, we join the search for extensions of this powerful tool beyond planarity by considering the h-framed graphs, a graph class that includes 1-planar, optimal 2-planar, and k-map graphs (for appropriate values of h). We establish a graph product structure theorem for h-framed graphs stating that the graphs in this class are subgraphs of the strong product of a path, of a planar graph of treewidth at most 3, and of a clique of size 3⌊ h/2 ⌋ +⌊ h/3 ⌋ -1. This allows us to improve over the previous structural theorems for 1-planar and k-map graphs. Our results constitute significant progress over the previous bounds on the queue number, non-repetitive chromatic number, and p-centered chromatic number of these graph classes, e.g., we lower the currently best upper bound on the queue number of 1-planar graphs and k-map graphs from 495 to 81 and from 32225k(k-3) to 61k, respectively. We also employ the product structure machinery to improve the current upper bounds of twin-width of planar and 1-planar graphs from 183 to 37, and from O(1) to 80, respectively. All our structural results are constructive and yield efficient algorithms to obtain the corresponding decompositions.Michael A. Bekos, Giordano Da Lozzo, Petr Hliněný, Michael Kaufmannwork_2nvonsl6svaidjkrnjopigafkyMon, 25 Apr 2022 00:00:00 GMTMatching minors in bipartite graphs
https://scholar.archive.org/work/ulk6jjkjnzh2hnh5ex5nky27a4
In this thesis we adapt fundamental parts of the Graph Minors series of Robertson and Seymour for the study of matching minors and investigate a connection to the study of directed graphs. We develope matching theoretic to established results of graph minor theory: We characterise the existence of a cross over a conformal cycle by means of a topological property. Furthermore, we develope a theory for perfect matching width, a width parameter for graphs with perfect matchings introduced by Norin. here we show that the disjoint alternating paths problem can be solved in polynomial time on graphs of bounded width. Moreover, we show that every bipartite graph with high perfect matching width must contain a large grid as a matching minor. Finally, we prove an analogue of the we known Flat Wall theorem and provide a qualitative description of all bipartite graphs which exclude a fixed matching minor.Sebastian Wiederrecht, Technische Universität Berlin, Stephan Kreutzerwork_ulk6jjkjnzh2hnh5ex5nky27a4Tue, 19 Apr 2022 00:00:00 GMTTwin-width VIII: delineation and win-wins
https://scholar.archive.org/work/pquptnv6a5ehjcpi2e4py2nxx4
We introduce the notion of delineation. A graph class 𝒞 is said delineated if for every hereditary closure 𝒟 of a subclass of 𝒞, it holds that 𝒟 has bounded twin-width if and only if 𝒟 is monadically dependent. An effective strengthening of delineation for a class 𝒞 implies that tractable FO model checking on 𝒞 is perfectly understood: On hereditary closures 𝒟 of subclasses of 𝒞, FO model checking is fixed-parameter tractable (FPT) exactly when 𝒟 has bounded twin-width. Ordered graphs [BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we show that segment graphs, directed path graphs, and visibility graphs of simple polygons are not delineated. In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA '21]. We show that K_t,t-free segment graphs, and axis-parallel H_t-free unit segment graphs have bounded twin-width, where H_t is the half-graph or ladder of height t. In contrast, axis-parallel H_4-free two-lengthed segment graphs have unbounded twin-width. Our new results, combined with the known FPT algorithm for FO model checking on graphs given with O(1)-sequences, lead to win-win arguments. For instance, we derive FPT algorithms for k-Ladder on visibility graphs of 1.5D terrains, and k-Independent Set on visibility graphs of simple polygons.Édouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, Noleen Köhler, Raul Lopes, Stéphan Thomasséwork_pquptnv6a5ehjcpi2e4py2nxx4Fri, 01 Apr 2022 00:00:00 GMTLossy Planarization: A Constant-Factor Approximate Kernelization for Planar Vertex Deletion
https://scholar.archive.org/work/377llva5ajasfplsbob5tng6gi
In the F-minor-free deletion problem we want to find a minimum vertex set in a given graph that intersects all minor models of graphs from the family F. The Vertex planarization problem is a special case of F-minor-free deletion for the family F = K_5, K_3,3. Whenever the family F contains at least one planar graph, then F-minor-free deletion is known to admit a constant-factor approximation algorithm and a polynomial kernelization [Fomin, Lokshtanov, Misra, and Saurabh, FOCS'12]. The Vertex planarization problem is arguably the simplest setting for which F does not contain a planar graph and the existence of a constant-factor approximation or a polynomial kernelization remains a major open problem. In this work we show that Vertex planarization admits an algorithm which is a combination of both approaches. Namely, we present a polynomial A-approximate kernelization, for some constant A > 1, based on the framework of lossy kernelization [Lokshtanov, Panolan, Ramanujan, and Saurabh, STOC'17]. Simply speaking, when given a graph G and integer k, we show how to compute a graph G' on poly(k) vertices so that any B-approximate solution to G' can be lifted to an (A*B)-approximate solution to G, as long as A*B*OPT(G) <= k. In order to achieve this, we develop a framework for sparsification of planar graphs which approximately preserves all separators and near-separators between subsets of the given terminal set. Our result yields an improvement over the state-of-art approximation algorithms for Vertex planarization. The problem admits a polynomial-time O(n^eps)-approximation algorithm, for any eps > 0, and a quasi-polynomial-time (log n)^O(1) approximation algorithm, both randomized [Kawarabayashi and Sidiropoulos, FOCS'17]. By pipelining these algorithms with our approximate kernelization, we improve the approximation factors to respectively O(OPT^eps) and (log OPT)^O(1).Bart M. P. Jansen, Michał Włodarczykwork_377llva5ajasfplsbob5tng6giFri, 04 Feb 2022 00:00:00 GMTRecent Advances in Positive-Instance Driven Graph Searching
https://scholar.archive.org/work/rmv3k2gc35fq3hxd6kdqq3i4cu
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.Max Bannach, Sebastian Berndtwork_rmv3k2gc35fq3hxd6kdqq3i4cuThu, 27 Jan 2022 00:00:00 GMTDeletion to Scattered Graph Classes II – Improved FPT Algorithms for Deletion to Pairs of Graph Classes
https://scholar.archive.org/work/24pjwql3lfbcrlnevvvsdcifj4
Let Π be a hereditary graph class. The problem of deletion to Π, takes as input a graph G and asks for a minimum number (or a fixed integer k) of vertices to be deleted from G so that the resulting graph belongs to Π. This is a well-studied problem in paradigms including approximation and parameterized complexity. Recently, the study of a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes, and the goal is to determine whether k vertices can be deleted from a given graph so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem is shown to be FPT as long as the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes is expressible in the counting monodic second order (CMSO) logic. While this was shown using some black box theorems, faster algorithms were shown when each of the hereditary graph classes has a finite forbidden set. In this paper, we do a deep dive on pairs of specific graph classes (Π_1, Π_2) in which we would like the connected components of the resulting graph to belong to, and design simpler and more efficient FPT algorithms. We design a general FPT algorithm and approximation algorithm for pairs of graph classes (possibly having infinite forbidden sets) satisfying certain conditions. These algorithms cover several pairs of popular graph classes. Our algorithm makes non-trivial use of the branching technique and as a black box, FPT algorithms for deletion to individual graph classes.Ashwin Jacob, Diptapriyo Majumdar, Venkatesh Ramanwork_24pjwql3lfbcrlnevvvsdcifj4Mon, 10 Jan 2022 00:00:00 GMTDeleting, Eliminating and Decomposing to Hereditary Classes Are All FPT-Equivalent
https://scholar.archive.org/work/izvuywq5wrdr5az7qxcittsq6m
For a graph class H, the graph parameters elimination distance to H (denoted by ed_ H) [Bulian and Dawar, Algorithmica, 2016], and H-treewidth (denoted by tw_ H) [Eiben et al. JCSS, 2021] aim to minimize the treedepth and treewidth, respectively, of the "torso" of the graph induced on a modulator to the graph class H. Here, the torso of a vertex set S in a graph G is the graph with vertex set S and an edge between two vertices u, v ∈ S if there is a path between u and v in G whose internal vertices all lie outside S. In this paper, we show that from the perspective of (non-uniform) fixed-parameter tractability (FPT), the three parameters described above give equally powerful parameterizations for every hereditary graph class H that satisfies mild additional conditions. In fact, we show that for every hereditary graph class H satisfying mild additional conditions, with the exception of tw_ H parameterized by ed_ H, for every pair of these parameters, computing one parameterized by itself or any of the others is FPT-equivalent to the standard vertex-deletion (to H) problem. As an example, we prove that an FPT algorithm for the vertex-deletion problem implies a non-uniform FPT algorithm for computing ed_ H and tw_ H. The conclusions of non-uniform FPT algorithms being somewhat unsatisfactory, we essentially prove that if H is hereditary, union-closed, CMSO-definable, and (a) the canonical equivalence relation (or any refinement thereof) for membership in the class can be efficiently computed, or (b) the class admits a "strong irrelevant vertex rule", then there exists a uniform FPT algorithm for ed_ H.Akanksha Agrawal, Lawqueen Kanesh, Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, Saket Saurabh, Meirav Zehaviwork_izvuywq5wrdr5az7qxcittsq6mFri, 07 Jan 2022 00:00:00 GMTEvaluation and Enumeration of Regular Simple Path and Trail Queries
https://scholar.archive.org/work/s65qs34zhrg45fns44745oshpi
Regular path queries (RPQs) are an essential component of graph query languages. Such queries consider a regular expression r and a directed edge-labeled graph G and search for paths in G for which the sequence of labels is in the language of r. In order to avoid having to consider infinitely many paths, some database engines restrict such paths to paths without repeated nodes or edges which are called simple paths or trails, respectively. Whereas arbitrary paths can be dealt with efficiently, simple paths and trails become computationally difficult already for very small RPQs. In this dissertation we investigate decision and enumeration problems concerning simple path and trail semantics. Evaluation Problem on Directed Graphs: Bagan, Bonifati, and Groz gave a trichotomy for the evaluation problem for simple paths when the RPQ is fixed. We complement their work by giving a similar trichotomy for the evaluation problem for trails and studying various characteristics of this class. We also study RPQs used in query logs and define a class of simple transitive expressions that is prominent in practice and for which we can prove dichotomies for the evaluation problem when the input language is not fixed, but used as a parameter. We observe that, even though simple path and trail semantics are intractable for RPQs in general, they are feasible for the vast majority of RPQs that are used in practice. At the heart of this study is a result of independent interest: the two disjoint paths problem in directed graphs is W[1]-hard if parameterized by the length of one of the two paths. Evaluation Problem on Undirected Graphs: While graph databases focus on directed graphs, there are edges which are naturally bidirectional, such as "sibling" or "married". Furthermore, database systems often allow to navigate an edge in its inverse direction (2RPQ), thus the study of the undirected setting gives us a better idea of what is possible. We are able to identify several tractable and intractable subclasses of regular languages when t [...]Tina Poppwork_s65qs34zhrg45fns44745oshpiCycle structure and colorings of directed graphs
https://scholar.archive.org/work/4oljppo7pnh3vlrxkkkho6zd4m
This thesis deals with problems from the theory of finite directed graphs. A directed graph (digraph for short) is a binary relation whose domain has finite size. With that digraphs can be seen as a very general way of representing (possibly asymmetric) relations between pairs from a finite set of objects. Undoubtedly, such a generality allows to encode many structures by digraphs. This works particularly well if important properties of the structure at hand can be expressed as relations or connections between objects. To give some selected examples, let us mention road networks, electricity networks, radio networks, the world wide web, circuits in electronic devices, or neural networks. A main focus of the thesis at hand is the investigation of properties of one of the most fundamental objects all over graph theory, the so-called cycle (sometimes also called circuit). A cycle in a graph is determined by a closed alternating sequence of cyclically connected vertices and edges. In a graph of finite size one will typically see loads of distinct cycles of various types. Therefore cycles constitute an important and recurring motive in almost all branches of graph theory, for instance, they play important roles in structural graph theory, in the theory of flows on directed networks, in theoretical characterizations of graph classes, as well as in the theory of graph colorings. Additionally, cycles play a decisive role in numerous algorithmic problems and their solutions, such as in the Traveling Salesman Problem, algorithms for finding a largest matching in a given graph, in the max-flow problem, and also in subprocedures such as Kruskal's algorithm for finding a minimum weight spanning tree. For those reasons, a substantial amount of research in graph theory has specialised on the structure of cycles in graphs. In the first major part of this thesis we deal with cycles which occur in directed graphs, and prove several necessary and sufficient theoretical conditions for the existence of cycles of certain types. Additi [...]Raphael Mario Steiner, Technische Universität Berlin, Stefan Felsnerwork_4oljppo7pnh3vlrxkkkho6zd4mThu, 30 Dec 2021 00:00:00 GMTIntroducing lop-kernels: a framework for kernelization lower bounds
https://scholar.archive.org/work/sdqe274ixfchtjem2e7ehhdi3i
In the Maximum Minimal Vertex Cover (MMVC) problem, we are given a graph G and a positive integer k, and the objective is to decide whether G contains a minimal vertex cover of size at least k. Motivated by the kernelization of MMVC with parameter k, our main contribution is to introduce a simple general framework to obtain kernelization lower bounds for a certain type of kernels for optimization problems, which we call lop-kernels. Informally, this type of kernels is required to preserve large optimal solutions in the reduced instance, and captures the vast majority of existing kernels in the literature. As a consequence of this framework, we show that the trivial quadratic kernel for MMVC is essentially optimal, answering a question of Boria et al. [Discret. Appl. Math. 2015], and that the known cubic kernel for Maximum Minimal Feedback Vertex Set is also essentially optimal. We present further applications for Tree Deletion Set and for Maximum Independent Set on K_t-free graphs. Back to the MMVC problem, given the (plausible) non-existence of subquadratic kernels for MMVC on general graphs, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdös-Hajnal property. Finally, we prove that MMVC does not admit polynomial kernels parameterized by the size of a minimum vertex cover of the input graph, even on bipartite graphs, unless NP⊆ coNP / poly.Júlio Araújo, Marin Bougeret, Victor A. Campos, Ignasi Sauwork_sdqe274ixfchtjem2e7ehhdi3iFri, 17 Dec 2021 00:00:00 GMTOn the scramble number of graphs
https://scholar.archive.org/work/tuojypxatvgkxixp4fnz32tofe
The scramble number of a graph is an invariant recently developed to aid in the study of divisorial gonality. In this paper we prove that scramble number is NP-hard to compute, also providing a proof that computing gonality is NP-hard even for simple graphs, as well as for metric graphs. We also provide general lower bounds for the scramble number of a Cartesian product of graphs, and apply these to compute gonality for many new families of product graphs.Marino Echavarria, Max Everett, Robin Huang, Liza Jacoby, Ralph Morrison, Ben Weberwork_tuojypxatvgkxixp4fnz32tofeTue, 07 Dec 2021 00:00:00 GMTClassifying grounded intersection graphs via ordered forbidden patterns
https://scholar.archive.org/work/tgjfj3husbegfgnui3vtvclgdq
It was noted already in the 90s that many classic graph classes, such as interval, chordal, and bipartite graphs, can be characterized by the existence of an ordering of the vertices avoiding some ordered subgraphs, called patterns. Very recently, all the classes corresponding to patterns on three vertices (including the ones mentioned above) have been listed, and proved to be efficiently recognizable. In contrast, very little is known about patterns on four vertices. One of the few graph classes characterized by a pattern on four vertices is the class of intersection graphs of rectangles that are said to be grounded on a line. This class appears naturally in the study of intersection graphs, and similar grounded classes have recently attracted a lot of attention. This paper contains three parts. First, we make a survey of grounded intersection graph classes, summarizing all the known inclusions between these various classes. Second, we show that the correspondence between a pattern on four vertices and grounded rectangle graphs is not an isolated phenomenon. We establish several other pattern characterizations for geometric classes, and show that the hierarchy of grounded intersection graph classes is tightly interleaved with the classes defined patterns on four vertices. We claim that forbidden patterns are a useful tool to classify grounded intersection graphs. Finally, we give an overview of the complexity of the recognition of classes defined by forbidden patterns on four vertices and list several interesting open problems.Laurent Feuilloley, Michel Habibwork_tgjfj3husbegfgnui3vtvclgdqMon, 06 Dec 2021 00:00:00 GMTDagstuhl Reports, Volume 11, Issue 6, July 2021, Complete Issue
https://scholar.archive.org/work/yqjei4bptngg7isjnkgk7byv7q
Dagstuhl Reports, Volume 11, Issue 6, June 2021, Complete Issuework_yqjei4bptngg7isjnkgk7byv7qWed, 01 Dec 2021 00:00:00 GMT