Discrete Mathematics Rooted Directed Path Graphs Are Leaf Powers
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A Forbidden Subgraph Characterization of Some Graph Classes Using Betweenness Axioms
A forbidden subgraph characterization of some graph classes using betweenness axioms Manoj Changat Department of Futures Studies University of Kerala Trivandrum - 695034, India e-mail: [email protected] Anandavally K Lakshmikuttyamma ∗ Department of Futures Studies University of Kerala, Trivandrum-695034, India e-mail: [email protected] Joseph Mathews Iztok Peterin C-GRAF, Department of Futures Studies University of Maribor University of Kerala, Trivandrum-695034, India FEECS, Smetanova 17, e-mail:[email protected] 2000 Maribor, Slovenia [email protected] Prasanth G. Narasimha-Shenoi Geetha Seethakuttyamma Department of Mathematics Department of Mathematics Government College, Chittur College of Engineering Palakkad - 678104, India Karungapally e-mail: [email protected] e-mail:[email protected] Simon Špacapan University of Maribor, FME, Smetanova 17, 2000 Maribor, Slovenia. e-mail: [email protected]. July 27, 2012 ∗ The work of this author (On leave from Mar Ivanios College, Trivandrum) is supported by the University Grants Commission (UGC), Govt. of India under the FIP scheme. 1 Abstract Let IG(x,y) and JG(x,y) be the geodesic and induced path interval between x and y in a connected graph G, respectively. The following three betweenness axioms are considered for a set V and R : V × V → 2V (i) x ∈ R(u,y),y ∈ R(x,v),x 6= y, |R(u,v)| > 2 ⇒ x ∈ R(u,v) (ii) x ∈ R(u,v) ⇒ R(u,x) ∩ R(x,v)= {x} (iii)x ∈ R(u,y),y ∈ R(x,v),x 6= y, ⇒ x ∈ R(u,v). We characterize the class of graphs for which IG satisfies (i), and the class for which JG satisfies (ii) and the class for which IG or JG satisfies (iii). -
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. -
(Eds.) Innovations in Bayesian Networks
Dawn E. Holmes and Lakhmi C. Jain (Eds.) InnovationsinBayesianNetworks Studies in Computational Intelligence,Volume 156 Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: Vol. 145. Mikhail Ju. Moshkov, Marcin Piliszczuk springer.com and Beata Zielosko Partial Covers, Reducts and Decision Rules in Rough Sets, 2008 Vol. 134. Ngoc Thanh Nguyen and Radoslaw Katarzyniak (Eds.) ISBN 978-3-540-69027-6 New Challenges in Applied Intelligence Technologies, 2008 Vol. 146. Fatos Xhafa and Ajith Abraham (Eds.) ISBN 978-3-540-79354-0 Metaheuristics for Scheduling in Distributed Computing Vol. 135. Hsinchun Chen and Christopher C.Yang (Eds.) Environments, 2008 Intelligence and Security Informatics, 2008 ISBN 978-3-540-69260-7 ISBN 978-3-540-69207-2 Vol. 147. Oliver Kramer Self-Adaptive Heuristics for Evolutionary Computation, 2008 Vol. 136. Carlos Cotta, Marc Sevaux ISBN 978-3-540-69280-5 and Kenneth S¨orensen (Eds.) Adaptive and Multilevel Metaheuristics, 2008 Vol. 148. Philipp Limbourg ISBN 978-3-540-79437-0 Dependability Modelling under Uncertainty, 2008 ISBN 978-3-540-69286-7 Vol. 137. Lakhmi C. Jain, Mika Sato-Ilic, Maria Virvou, George A. Tsihrintzis,Valentina Emilia Balas Vol. 149. Roger Lee (Ed.) and Canicious Abeynayake (Eds.) Software Engineering, Artificial Intelligence, Networking and Computational Intelligence Paradigms, 2008 Parallel/Distributed Computing, 2008 ISBN 978-3-540-79473-8 ISBN 978-3-540-70559-8 Vol. 138. Bruno Apolloni,Witold Pedrycz, Simone Bassis Vol. 150. Roger Lee (Ed.) and Dario Malchiodi Software Engineering Research, Management and The Puzzle of Granular Computing, 2008 Applications, 2008 ISBN 978-3-540-79863-7 ISBN 978-3-540-70774-5 Vol. -
Graph Isomorphism Completeness for Chordal Bipartite Graphs and Strongly Chordal Graphs
Graph Isomorphism Completeness for Chordal bipartite graphs and Strongly Chordal Graphs Ryuhei Uehara a Seinosuke Toda b Takayuki Nagoya c aNatural Science Faculty, Komazawa University. 1 bDepartment of Computer Science and System Analysis, College of Humanities and Sciences, Nihon University. cDepartment of Mathematical Sciences, Tokyo Denki University. Abstract This paper deals with the graph isomorphism (GI) problem for two graph classes: chordal bipartite graphs and strongly chordal graphs. It is known that GI problem is GI complete even for some special graph classes including regular graphs, bipar- tite graphs, chordal graphs, comparability graphs, split graphs, and k-trees with unbounded k. On the other side, the relative complexity of the GI problem for the above classes was unknown. We prove that deciding isomorphism of the classes are GI complete. Key words: Graph isomorphism problem, Graph isomorphism complete, Strongly chordal graphs, Chordal bipartite graphs 1 Introduction The graph isomorphism (GI) problem is a well-known problem and exploring its precise complexity has become an important open question in computa- tional complexity theory three decades ago (see (Kar72)). Although the prob- lem is trivially in NP, the problem is not known to be in P and not known to be NP-complete either (see (RC77; KST93)). It is very unlikely that the Email addresses: [email protected] (Ryuhei Uehara), [email protected] (Seinosuke Toda), [email protected] (Takayuki Nagoya). 1 This work was done while the author was visiting University of Waterloo. Preprint submitted to Elsevier Science 26 May 2004 GI problem is NP-complete. -
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. -
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. -
Graph Convexity and Vertex Orderings by Rachel Jean Selma Anderson B.Sc., Mcgill University, 2002 B.Ed., University of Victoria
Graph Convexity and Vertex Orderings by Rachel Jean Selma Anderson B.Sc., McGill University, 2002 B.Ed., University of Victoria, 2007 AThesisSubmittedinPartialFulfillmentofthe Requirements for the Degree of MASTER OF SCIENCE in the Department of Mathematics and Statistics c Rachel Jean Selma Anderson, 2014 University of Victoria All rights reserved. This thesis may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii Graph Convexity and Vertex Orderings by Rachel Jean Selma Anderson B.Sc., McGill University, 2002 B.Ed., University of Victoria, 2007 Supervisory Committee Dr. Ortrud Oellermann, Co-supervisor (Department of Mathematics and Statistics, University of Winnipeg) Dr. Gary MacGillivray, Co-supervisor (Department of Mathematics and Statistics, University of Victoria) iii Supervisory Committee Dr. Ortrud Oellermann, Co-supervisor (Department of Mathematics and Statistics, University of Winnipeg) Dr. Gary MacGillivray, Co-supervisor (Department of Mathematics and Statistics, University of Victoria) ABSTRACT In discrete mathematics, a convex space is an ordered pair (V, )where is a M M family of subsets of a finite set V ,suchthat: , V ,and is closed under ∅∈M ∈M M intersection. The elements of are called convex sets.ForasetS V ,theconvex M ⊆ hull of S is the smallest convex set that contains S.Apointx of a convex set X is an extreme point of X if X x is also convex. A convex space (V, )withtheproperty \{ } M that every convex set is the convex hull of its extreme points is called a convex geome- try.AgraphG has a P-elimination ordering if an ordering v1,v2,...,vn of the vertices exists such that vi has property P in the graph induced by vertices vi,vi+1,...,vn for all i =1, 2,...,n. -
Closest 4-Leaf Power Is Fixed-Parameter Tractable ⋆
Discrete Applied Mathematics, to appear, 2008 Closest 4-Leaf Power is Fixed-Parameter Tractable ⋆ Michael Dom, Jiong Guo, Falk H¨uffner, Rolf Niedermeier Abstract The NP-complete Closest 4-Leaf Power problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree—the 4-leaf root—with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing previous work on Closest 2-Leaf Power and Closest 3-Leaf Power, we give the first algorithmic result for Closest 4-Leaf Power, showing that Closest 4-Leaf Power is fixed-parameter tractable with respect to the parameter r. Key words: fixed-parameter tractability, graph algorithm, graph modification, graph power, leaf power, forbidden subgraph characterization 1 Introduction Graph powers form a classical concept in graph theory, and the rich literature dates back to the sixties of the previous century (see [5, Sect. 10.6] and [27] for surveys). The k-power of an undirected graph G = (V, E) is the undirected graph Gk = (V, E′) with (u, v) ∈ E′ iff there is a path of length at most k between u and v in G. We say G is the k-root of Gk. It is NP-complete to decide whether a given graph is a 2-power (square) [30]. -
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]. -
A Polynomial Kernel for 3-Leaf Power Deletion
A Polynomial Kernel for 3-Leaf Power Deletion Jungho Ahn Department of Mathematical Sciences, KAIST, Daejeon, South Korea Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea [email protected] Eduard Eiben Department of Computer Science, Royal Holloway, University of London, Egham, UK [email protected] O-joung Kwon Department of Mathematics, Incheon National University, South Korea Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea [email protected] Sang-il Oum Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea Department of Mathematical Sciences, KAIST, Daejeon, South Korea [email protected] Abstract For a non-negative integer `, a graph G is an `-leaf power of a tree T if V pGq is equal to the set of leaves of T , and distinct vertices v and w of G are adjacent if and only if the distance between v and w in T is at most `. Given a graph G, 3-Leaf Power Deletion asks whether there is a set S Ď V pGq of size at most k such that GzS is a 3-leaf power of some tree T . We provide a polynomial kernel for this problem. More specifically, we present a polynomial-time algorithm for an input instance pG, kq to output an equivalent instance pG1, k1q such that k1 ¤ k and G1 has at most Opk14q vertices. 2012 ACM Subject Classification Theory of computation Ñ Design and analysis of algorithms; Theory of computation Ñ Graph algorithms analysis; Theory of computation Ñ Parameterized complexity and exact algorithms Keywords and phrases `-leaf power, parameterized algorithms, kernelization Digital Object Identifier 10.4230/LIPIcs.MFCS.2020.5 Related Version A full version of this paper is available at https://arxiv.org/abs/1911.04249. -
Characterization and Recognition of Some Opposition and Coalition
Characterization and recognition of some opposition and coalition graph classes Van Bang Le Universit¨at Rostock, Institut fur¨ Informatik, Rostock, Germany Thomas Podelleck Universit¨at Rostock, Institut fur¨ Informatik, Rostock, Germany Abstract A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coali- tion graphs have been introduced and investigated in connection to perfect graphs. Recognizing and characterizing opposition and coalition graphs are long-standing open problems. This paper gives characterizations for opposi- tion graphs and coalition graphs on some restricted graph classes. Implicit in our arguments are polynomial time recognition algorithms for these graphs. We also give a good characterization for the so-called generalized opposition graphs. Keywords: Perfectly orderable graph, one-in-one-out graph, opposition graph, coalition graph, perfect graph 2010 MSC: 05C75, 05C17, 05C62 arXiv:1507.00557v1 [cs.DM] 2 Jul 2015 1. Introduction and preliminaries Chv´atal [4] proposed to call a linear order < on the vertex set of an undirected graph G perfect if the greedy coloring algorithm applied to each induced subgraph H of G gives an optimal coloring of H: Consider the vertices of H sequentially by following the order < and assign to each vertex Email addresses: [email protected] (Van Bang Le), [email protected] (Thomas Podelleck) To appear in Discrete Appl. Math. November 8, 2017 v the smallest color not used on any neighbor u of v, u < v. A graph is perfectly orderable if it admits a perfect order.