Intersection Graphs of Paths in a Tree
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Mutant Knots and Intersection Graphs 1 Introduction
Mutant knots and intersection graphs S. V. CHMUTOV S. K. LANDO We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. Conversely, if we have a weight system depending only on the intersection graphs of chord diagrams, then the composition of such a weight system with the Kontsevich invariant determines a knot invariant that does not distinguish mutant knots. Thus, an equivalence between finite order invariants not distinguishing mutants and weight systems depending on intersections graphs only is established. We discuss relationship between our results and certain Lie algebra weight systems. 57M15; 57M25 1 Introduction Below, we use standard notions of the theory of finite order, or Vassiliev, invariants of knots in 3-space; their definitions can be found, for example, in [6] or [14], and we recall them briefly in Section 2. All knots are assumed to be oriented. Two knots are said to be mutant if they differ by a rotation of a tangle with four endpoints about either a vertical axis, or a horizontal axis, or an axis perpendicular to the paper. If necessary, the orientation inside the tangle may be replaced by the opposite one. Here is a famous example of mutant knots, the Conway (11n34) knot C of genus 3, and Kinoshita–Terasaka (11n42) knot KT of genus 2 (see [1]). C = KT = Note that the change of the orientation of a knot can be achieved by a mutation in the complement to a trivial tangle. -
Exclusive Graph Searching Lélia Blin, Janna Burman, Nicolas Nisse
Exclusive Graph Searching Lélia Blin, Janna Burman, Nicolas Nisse To cite this version: Lélia Blin, Janna Burman, Nicolas Nisse. Exclusive Graph Searching. Algorithmica, Springer Verlag, 2017, 77 (3), pp.942-969. 10.1007/s00453-016-0124-0. hal-01266492 HAL Id: hal-01266492 https://hal.archives-ouvertes.fr/hal-01266492 Submitted on 2 Feb 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Exclusive Graph Searching∗ L´eliaBlin Sorbonne Universit´es, UPMC Univ Paris 06, CNRS, Universit´ed'Evry-Val-d'Essonne. LIP6 UMR 7606, 4 place Jussieu 75005, Paris, France [email protected] Janna Burman LRI, Universit´eParis Sud, CNRS, UMR-8623, France. [email protected] Nicolas Nisse Inria, France. Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, Sophia Antipolis, France. [email protected] February 2, 2016 Abstract 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. -
Discrete Mathematics Rooted Directed Path Graphs Are Leaf Powers
Discrete Mathematics 310 (2010) 897–910 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Rooted directed path graphs are leaf powers Andreas Brandstädt a, Christian Hundt a, Federico Mancini b, Peter Wagner a a Institut für Informatik, Universität Rostock, D-18051, Germany b Department of Informatics, University of Bergen, N-5020 Bergen, Norway article info a b s t r a c t Article history: Leaf powers are a graph class which has been introduced to model the problem of Received 11 March 2009 reconstructing phylogenetic trees. A graph G D .V ; E/ is called k-leaf power if it admits a Received in revised form 2 September 2009 k-leaf root, i.e., a tree T with leaves V such that uv is an edge in G if and only if the distance Accepted 13 October 2009 between u and v in T is at most k. Moroever, a graph is simply called leaf power if it is a Available online 30 October 2009 k-leaf power for some k 2 N. This paper characterizes leaf powers in terms of their relation to several other known graph classes. It also addresses the problem of deciding whether a Keywords: given graph is a k-leaf power. Leaf powers Leaf roots We show that the class of leaf powers coincides with fixed tolerance NeST graphs, a Graph class inclusions well-known graph class with absolutely different motivations. After this, we provide the Strongly chordal graphs largest currently known proper subclass of leaf powers, i.e, the class of rooted directed path Fixed tolerance NeST graphs graphs. -
Representations of Edge Intersection Graphs of Paths in a Tree Martin Charles Golumbic, Marina Lipshteyn, Michal Stern
Representations of Edge Intersection Graphs of Paths in a Tree Martin Charles Golumbic, Marina Lipshteyn, Michal Stern To cite this version: Martin Charles Golumbic, Marina Lipshteyn, Michal Stern. Representations of Edge Intersection Graphs of Paths in a Tree. 2005 European Conference on Combinatorics, Graph Theory and Appli- cations (EuroComb ’05), 2005, Berlin, Germany. pp.87-92. hal-01184396 HAL Id: hal-01184396 https://hal.inria.fr/hal-01184396 Submitted on 14 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EuroComb 2005 DMTCS proc. AE, 2005, 87–92 Representations of Edge Intersection Graphs of Paths in a Tree Martin Charles Golumbic1,† Marina Lipshteyn1 and Michal Stern1 1Caesarea Rothschild Institute, University of Haifa, Haifa, Israel Let P be a collection of nontrivial simple paths in a tree T . The edge intersection graph of P, denoted by EP T (P), has vertex set that corresponds to the members of P, and two vertices are joined by an edge if the corresponding members of P share a common edge in T . An undirected graph G is called an edge intersection graph of paths in a tree, if G = EP T (P) for some P and T . -
NP-Complete and Cliques
CSC373— Algorithm Design, Analysis, and Complexity — Spring 2018 Tutorial Exercise 10: Cliques and Intersection Graphs for Intervals 1. Clique. Given an undirected graph G = (V,E) a clique (pronounced “cleek” in Canadian, heh?) is a subset of vertices K ⊆ V such that, for every pair of distinct vertices u,v ∈ K, the edge (u,v) is in E. See Clique, Graph Theory, Wikipedia. A second concept that will be useful is the notion of a complement graph Gc. We say Gc is the complement (graph) of G = (V,E) iff Gc = (V,Ec), where Ec = {(u,v) | u,v ∈ V,u 6= v, and (u,v) ∈/ E} (see Complement Graph, Wikipedia). That is, the complement graph Gc is the graph over the same set of vertices, but it contains all (and only) the edges that are not in E. Finally, a clique K ⊂ V of G =(V,E) is said to be a maximal clique iff there is no superset W such that K ⊂ W ⊆ V and W is a clique of G. Consider the decision problem: Clique: Given an undirected graph and an integer k, does there exist a clique K of G with |K|≥ k? Clearly, Clique is in NP. We wish to show that Clique is NP-complete. To do this, it is convenient to first note that the independent set problem IndepSet(G, k) is very closely related to the Clique problem for the complement graph, i.e., Clique(Gc,k). Use this strategy to show: IndepSet ≡p Clique. (1) 2. Consider the interval scheduling problem we started this course with (which is also revisited in Assignment 3, Question 1). -
Compact Representation of Graphs with Small Bandwidth and Treedepth
Compact Representation of Graphs with Small Bandwidth and Treedepth Shahin Kamali University of Manitoba Winnipeg, MB, R3T 2N2, Canada [email protected] Abstract We consider the problem of compact representation of graphs with small bandwidth as well as graphs with small treedepth. These parameters capture structural properties of graphs that come in useful in certain applications. We present simple navigation oracles that support degree and adjacency queries in constant time and neighborhood query in constant time per neighbor. For graphs of bandwidth k, our oracle takesp (k + dlog 2ke)n + o(kn) bits. By way of an enumeration technique, we show that (k − 5 k − 4)n − O(k2) bits are required to encode a graph of bandwidth k and size n. For graphs of treedepth k, our oracle takes (k + dlog ke + 2)n + o(kn) bits. We present a lower bound that certifies our oracle is succinct for certain values of k 2 o(n). 1 Introduction Graphs are among the most relevant ways to model relationships between objects. Given the ever-growing number of objects to model, it is increasingly important to store the underlying graphs in a compact manner in order to facilitate designing efficient algorithmic solutions. One simple way to represent graphs is to consider them as random objects without any particular structure. There are two issues, however, with such general approach. First, many computational problems are NP-hard on general graphs and often remain hard to approximate. Second, random graphs are highly incompressible and cannot be stored compactly [1]. At the same time, graphs that arise in practice often have combinatorial structures that can -and should- be exploited to provide space-efficient representations. -
On Intersection Graphs of Graphs and Hypergraphs: a Survey
Intersection Graphs of Graphs and Hypergraphs: A Survey Ranjan N. Naika aDepartment of Mathematics, Lincoln University, PA, USA [email protected] Abstract The survey is devoted to the developmental milestones on the characterizations of intersection graphs of graphs and hypergraphs. The theory of intersection graphs of graphs and hypergraphs has been a classical topic in the theory of special graphs. To conclude, at the end, we have listed some open problems posed by various authors whose work has contributed to this survey and also the new trends coming out of intersection graphs of hyeprgraphs. Keywords: Hypergraphs, Intersection graphs, Line graphs, Representative graphs, Derived graphs, Algorithms (ALG),, Forbidden induced subgraphs (FIS), Krausz partitions, Eigen values Mathematics Subject Classification : 06C62, 05C65, 05C75, 05C85, 05C69 1. Introduction We follow the terminology of Berge, C. [3] and [4]. This survey does not address intersection graphs of other types of graphs such as interval graphs etc. An introduction of intersection graphs of interval graphs etc. are available in Pal [40]. A graph or a hypergraph is a pair (V, E), where V is the vertex set and E is the edge set a family of nonempty subsets of V. Two edges of a hypergraph are l -intersecting if they share at least l common vertices. This concept was studied in [6] and [18] by Bermond, Heydemann and Sotteau. A hypergraph H is called a k-uniform hypergraph if its edges have k number of vertices. A hypergraph is linear if any two edges have at most one common vertex. A 2-uniform linear hypergraph is called a graph or a linear graph. -
Branch-Depth: Generalizing Tree-Depth of Graphs
Branch-depth: Generalizing tree-depth of graphs ∗1 †‡23 34 Matt DeVos , O-joung Kwon , and Sang-il Oum† 1Department of Mathematics, Simon Fraser University, Burnaby, Canada 2Department of Mathematics, Incheon National University, Incheon, Korea 3Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, Korea 4Department of Mathematical Sciences, KAIST, Daejeon, Korea [email protected], [email protected], [email protected] November 5, 2020 Abstract We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the no- tions of tree-depth and shrub-depth of graphs as follows. For a graph G = (V, E) and a subset A of E we let λG(A) be the number of vertices incident with an edge in A and an edge in E A. For a subset X of V , \ let ρG(X) be the rank of the adjacency matrix between X and V X over the binary field. We prove that a class of graphs has bounded\ tree-depth if and only if the corresponding class of functions λG has arXiv:1903.11988v2 [math.CO] 4 Nov 2020 bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree- depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by restriction. -
An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling
An Algorithmic Study of Fully Dynamic Independent Sets for Map Labeling Sujoy Bhore Algorithms and Complexity Group, TU Wien, Austria [email protected] Guangping Li Algorithms and Complexity Group, TU Wien, Austria [email protected] Martin Nöllenburg Algorithms and Complexity Group, TU Wien, Austria [email protected] Abstract Map labeling is a classical problem in cartography and geographic information systems (GIS) that asks to place labels for area, line, and point features, with the goal to select and place the maximum number of independent, i.e., overlap-free, labels. A practically interesting case is point labeling with axis-parallel rectangular labels of common size. In a fully dynamic setting, at each time step, either a new label appears or an existing label disappears. Then, the challenge is to maintain a maximum cardinality subset of pairwise independent labels with sub-linear update time. Motivated by this, we study the maximal independent set (MIS) and maximum independent set (Max-IS) problems on fully dynamic (insertion/deletion model) sets of axis-parallel rectangles of two types – (i) uniform height and width and (ii) uniform height and arbitrary width; both settings can be modeled as rectangle intersection graphs. We present the first deterministic algorithm for maintaining a MIS (and thus a 4-approximate Max-IS) of a dynamic set of uniform rectangles with amortized sub-logarithmic update time. This breaks the natural barrier of Ω(∆) update time (where ∆ is the maximum degree in the graph) for vertex updates presented by Assadi et al. (STOC 2018). We continue by investigating Max-IS and provide a series of deterministic dynamic approximation schemes. -
An Efficient Reachability Indexing Scheme for Large Directed Graphs
1 Path-Tree: An Efficient Reachability Indexing Scheme for Large Directed Graphs RUOMING JIN, Kent State University NING RUAN, Kent State University YANG XIANG, The Ohio State University HAIXUN WANG, Microsoft Research Asia Reachability query is one of the fundamental queries in graph database. The main idea behind answering reachability queries is to assign vertices with certain labels such that the reachability between any two vertices can be determined by the labeling information. Though several approaches have been proposed for building these reachability labels, it remains open issues on how to handle increasingly large number of vertices in real world graphs, and how to find the best tradeoff among the labeling size, the query answering time, and the construction time. In this paper, we introduce a novel graph structure, referred to as path- tree, to help labeling very large graphs. The path-tree cover is a spanning subgraph of G in a tree shape. We show path-tree can be generalized to chain-tree which theoretically can has smaller labeling cost. On top of path-tree and chain-tree index, we also introduce a new compression scheme which groups vertices with similar labels together to further reduce the labeling size. In addition, we also propose an efficient incremental update algorithm for dynamic index maintenance. Finally, we demonstrate both analytically and empirically the effectiveness and efficiency of our new approaches. Categories and Subject Descriptors: H.2.8 [Database management]: Database Applications—graph index- ing and querying General Terms: Performance Additional Key Words and Phrases: Graph indexing, reachability queries, transitive closure, path-tree cover, maximal directed spanning tree ACM Reference Format: Jin, R., Ruan, N., Xiang, Y., and Wang, H. -
Cop Number of Graphs Without Long Holes
Cop number of graphs without long holes Vaidy Sivaraman January 3, 2020 Abstract A hole in a graph is an induced cycle of length at least 4. We give a simple winning strategy for t − 3 cops to capture a robber in the game of cops and robbers played in a graph that does not contain a hole of length at least t. This strengthens a theorem of Joret-Kaminski-Theis, who proved that t − 2 cops have a winning strategy in such graphs. As a consequence of our bound, we also give an inequality relating the cop number and the Dilworth number of a graph. The game of cops and robbers is played on a finite simple graph G. There are k cops and a single robber. Each of the cops chooses a vertex to start, and then the robber chooses a vertex. And then they alternate turns starting with the cop. In the turn of cops, each cop either stays on the vertex or moves to a neighboring vertex. In the robber’s turn, he stays on the same vertex or moves to a neighboring vertex. The cops win if at any point in time, one of the cops lands on the robber. The robber wins if he can avoid being captured. The cop number of G, denoted c(G), is the minimum number of cops needed so that the cops have a winning strategy in G. The question of what makes a graph to have high cop number is not clearly understood. Some fundamental results were proved in [1] and [9]. -
Tree-Depth and the Formula Complexity of Subgraph Isomorphism
Electronic Colloquium on Computational Complexity, Report No. 61 (2020) Tree-depth and the Formula Complexity of Subgraph Isomorphism Deepanshu Kush Benjamin Rossman University of Toronto Duke University April 28, 2020 Abstract For a fixed \pattern" graph G, the colored G-subgraph isomorphism problem (denoted SUB(G)) asks, given an n-vertex graph H and a coloring V (H) ! V (G), whether H contains a properly colored copy of G. The complexity of this problem is tied to parameterized versions of P =? NP and L =? NL, among other questions. An overarching goal is to understand the complexity of SUB(G), under different computational models, in terms of natural invariants of the pattern graph G. In this paper, we establish a close relationship between the formula complexity of SUB(G) and an invariant known as tree-depth (denoted td(G)). SUB(G) is known to be solvable by monotone AC 0 1=3 formulas of size O(ntd(G)). Our main result is an nΩ(e td(G) ) lower bound for formulas that are monotone or have sub-logarithmic depth. This complements a lower bound of Li, Razborov and Rossman [8] relating tree-width and AC 0 circuit size. As a corollary, it implies a stronger homomorphism preservation theorem for first-order logic on finite structures [14]. The technical core of this result is an nΩ(k) lower bound in the special case where G is a complete binary tree of height k, which we establish using the pathset framework introduced in [15]. (The lower bound for general patterns follows via a recent excluded-minor characterization of tree-depth [4, 6].) Additional results of this paper extend the pathset framework and improve upon both, the best known upper and lower bounds on the average-case formula size of SUB(G) when G is a path.