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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). -
1 Vertex Connectivity 2 Edge Connectivity 3 Biconnectivity
1 Vertex Connectivity So far we've talked about connectivity for undirected graphs and weak and strong connec- tivity for directed graphs. For undirected graphs, we're going to now somewhat generalize the concept of connectedness in terms of network robustness. Essentially, given a graph, we may want to answer the question of how many vertices or edges must be removed in order to disconnect the graph; i.e., break it up into multiple components. Formally, for a connected graph G, a set of vertices S ⊆ V (G) is a separating set if subgraph G − S has more than one component or is only a single vertex. The set S is also called a vertex separator or a vertex cut. The connectivity of G, κ(G), is the minimum size of any S ⊆ V (G) such that G − S is disconnected or has a single vertex; such an S would be called a minimum separator. We say that G is k-connected if κ(G) ≥ k. 2 Edge Connectivity We have similar concepts for edges. For a connected graph G, a set of edges F ⊆ E(G) is a disconnecting set if G − F has more than one component. If G − F has two components, F is also called an edge cut. The edge-connectivity if G, κ0(G), is the minimum size of any F ⊆ E(G) such that G − F is disconnected; such an F would be called a minimum cut.A bond is a minimal non-empty edge cut; note that a bond is not necessarily a minimum cut. -
CLRS B.4 Graph Theory Definitions Unit 1: DFS Informally, a Graph
CLRS B.4 Graph Theory Definitions Unit 1: DFS informally, a graph consists of “vertices” joined together by “edges,” e.g.,: example graph G0: 1 ···················•······························· ····························· ····························· ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ············· ···· ···· ·············· 2•···· ···· ···· ··· •· 3 ···· ···· ···· ···· ···· ··· ···· ···· ···· ···· ···· ···· ···· ···· ···· ···· ······· ······· ······· ······· ···· ···· ···· ··· ···· ···· ···· ···· ···· ··· ···· ···· ···· ···· ···· ···· ···· ···· ···· ···· ··············· ···· ···· ··············· 4•························· ···· ···· ························· • 5 ························· ···· ···· ························· ························· ···· ···· ························· ························· ···· ···· ························· ····························· ····························· ···················•································ 6 formally a graph is a pair (V, E) where V is a finite set of elements, called vertices E is a finite set of pairs of vertices, called edges if H is a graph, we can denote its vertex & edge sets as V (H) & E(H) respectively if the pairs of E are unordered, the graph is undirected if the pairs of E are ordered the graph is directed, or a digraph two vertices joined by an edge -
Efficient Multicore Algorithms for Identifying Biconnected Components
International Journal of Networking and Computing { www.ijnc.org ISSN 2185-2839 (print) ISSN 2185-2847 (online) Volume 6, Number 1, pages 87{106, January 2016 Efficient Multicore Algorithms For Identifying Biconnected Components1 Meher Chaitanya [email protected] and Kishore Kothapalli [email protected] International Institute of Information Technology Hyderabad, Gachibowli, India 500 032. Received: July 30, 2015 Revised: October 26, 2015 Accepted: December 1, 2015 Communicated by Akihiro Fujiwara Abstract In this paper we design and implement an algorithm for finding the biconnected components of a given graph. Our algorithm is based on experimental evidence that finding the bridges of a graph is usually easier and faster in the parallel setting. We use this property to first decompose the graph into independent and maximal 2-edge-connected subgraphs. To identify the articulation points in these 2-edge connected subgraphs, we again convert this into a problem of finding the bridges on an auxiliary graph. It is interesting to note that during the conversion process, the size of the graph may increase. However, we show that this small increase in size and the run time is offset by the consideration that finding bridges is easier in a parallel setting. We implement our algorithm on an Intel i7 980X CPU running 12 threads. We show that our algorithm is on average 2.45x faster than the best known current algorithms implemented on the same platform. Finally, we extend our approach to dense graphs by applying the sparsification technique suggested by Cong and Bader in [7]. Keywords: Biconnected components, Least common ancestor, 2-edge connected components, Articulation points 1 Introduction The biconnected components of a given graph are its maximal 2-connected subgraphs. -
(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. -
On Computing Longest Paths in Small Graph Classes
On Computing Longest Paths in Small Graph Classes Ryuhei Uehara∗ Yushi Uno† July 28, 2005 Abstract The longest path problem is to find a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. For a tree, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for weighted trees, block graphs, and cacti. We next show that the longest path problem can be solved efficiently on some graph classes that have natural interval representations. Keywords: efficient algorithms, graph classes, longest path problem. 1 Introduction The Hamiltonian path problem is one of the most well known NP-hard problem, and there are numerous applications of the problems [17]. For such an intractable problem, there are two major approaches; approximation algorithms [20, 2, 35] and algorithms with parameterized complexity analyses [15]. In both approaches, we have to change the decision problem to the optimization problem. Therefore the longest path problem is one of the basic problems from the viewpoint of combinatorial optimization. From the practical point of view, it is also very natural approach to try to find a longest path in a given graph, even if it does not have a Hamiltonian path. However, finding a longest path seems to be more difficult than determining whether the given graph has a Hamiltonian path or not. -
Strongly Connected Components and Biconnected Components
Strongly Connected Components and Biconnected Components Daniel Wisdom 27 January 2017 1 2 3 4 5 6 7 8 Figure 1: A directed graph. Credit: Crash Course Coding Companion. 1 Strongly Connected Components A strongly connected component (SCC) is a set of vertices where every vertex can reach every other vertex. In an undirected graph the SCCs are just the groups of connected vertices. We can find them using a simple DFS. This works because, in an undirected graph, u can reach v if and only if v can reach u. This is not true in a directed graph, which makes finding the SCCs harder. More on that later. 2 Kernel graph We can travel between any two nodes in the same strongly connected component. So what if we replaced each strongly connected component with a single node? This would give us what we call the kernel graph, which describes the edges between strongly connected components. Note that the kernel graph is a directed acyclic graph because any cycles would constitute an SCC, so the cycle would be compressed into a kernel. 3 1,2,5,6 4,7 8 Figure 2: Kernel graph of the above directed graph. 3 Tarjan's SCC Algorithm In a DFS traversal of the graph every SCC will be a subtree of the DFS tree. This subtree is not necessarily proper, meaning that it may be missing some branches which are their own SCCs. This means that if we can identify every node that is the root of its SCC in the DFS, we can enumerate all the SCCs. -
Vertex Deletion Problems on Chordal Graphs∗†
Vertex Deletion Problems on Chordal Graphs∗† Yixin Cao1, Yuping Ke2, Yota Otachi3, and Jie You4 1 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 2 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 3 Faculty of Advanced Science and Technology, Kumamoto University, Kumamoto, Japan [email protected] 4 School of Information Science and Engineering, Central South University and Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] Abstract Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yan- nakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete. 1998 ACM Subject Classification F.2.2 Analysis of Algorithms and Problem Complexity, G.2.2 Graph Theory Keywords and phrases vertex deletion problem, maximum subgraph, chordal graph, (unit) in- terval graph, split graph, hereditary property, NP-complete, polynomial-time algorithm Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2017.22 1 Introduction Generally speaking, a vertex deletion problem asks to transform an input graph to a graph in a certain class by deleting a minimum number of vertices. -
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. -
Characterizing Simultaneous Embedding with Fixed Edges
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by computer science publication server Characterizing Simultaneous Embedding with Fixed Edges J. Joseph Fowler1, Michael J¨unger2, Stephen G. Kobourov1, and Michael Schulz2 1 University of Arizona, USA {jfowler,kobourov}@cs.arizona.edu ⋆ 2 University of Cologne, Germany {mjuenger,schulz}@informatik.uni-koeln.de ⋆⋆ Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same Jordan curve in the simultaneous drawings. While any number of planar graphs have a simultaneous embedding without fixed edges, determining which graphs always share a simultaneous embedding with fixed edges (SEFE) has been open. We partially close this problem by giving a necessary condition to determine when pairs of graphs have a SEFE. 1 Introduction In many practical applications including the visualization of large graphs and very-large-scale in- tegration (VLSI) of circuits on the same chip, edge crossings are undesirable. A single vertex set can suffice in which multiple edge sets correspond to different edge colors or circuit layers. While the union of any pair of edge sets may be nonplanar, a planar drawing of each layer may still be possible, as crossings between edges of distinct edge sets is permitted. This corresponds to the problem of simultaneous embedding (SE). Without restrictions on the types of edges used, this problem is trivial since any number of planar graphs can be drawn on the same fixed set of vertex locations [6]. -
Near-Linear Time Constant-Factor Approximation Algorithm for Branch-Decomposition of Planar Graphs
Near-Linear Time Constant-Factor Approximation Algorithm for Branch-Decomposition of Planar Graphs1 Qian-Ping Gu and Gengchun Xu School of Computing Science, Simon Fraser University Burnaby BC Canada V5A1S6 [email protected],[email protected] Abstract: We give an algorithm which for an input planar graph G of n vertices and integer k, in min{O(n log3 n),O(nk2)} time either constructs a branch-decomposition of G k+1 with width at most (2 + δ)k, δ > 0 is a constant, or a (k + 1) × ⌈ 2 ⌉ cylinder minor of G implying bw(G) >k, bw(G) is the branchwidth of G. This is the first O˜(n) time constant- factor approximation for branchwidth/treewidth and largest grid/cylinder minors of planar graphs and improves the previous min{O(n1+ǫ),O(nk2)} (ǫ> 0 is a constant) time constant- factor approximations. For a planar graph G and k = bw(G), a branch-decomposition of g k width at most (2 + δ)k and a g × 2 cylinder/grid minor with g = β , β > 2 is constant, can be computed by our algorithm in min{O(n log3 n log k),O(nk2 log k)} time. Key words: Branch-/tree-decompositions, grid minor, planar graphs, approximation algo- rithm. 1 Introduction The notions of branchwidth and branch-decomposition introduced by Robertson and Sey- mour [31] in relation to the notions of treewidth and tree-decomposition have important algorithmic applications. The branchwidth bw(G) and the treewidth tw(G) of graph G 3 are linearly related: max{bw(G), 2} ≤ tw(G)+1 ≤ max{⌊ 2 bw(G)⌋, 2} for every G with more than one edge, and there are simple translations between branch-decompositions and tree-decompositions that meet the linear relations [31]. -
Introducing Subclasses of Basic Chordal Graphs
Introducing subclasses of basic chordal graphs Pablo De Caria 1 Marisa Gutierrez 2 CONICET/ Universidad Nacional de La Plata, Argentina Abstract Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. In this work, we consider some subclasses of basic chordal graphs, like hereditary basic chordal graphs, basic DV and basic RDV graphs, we characterize them and we find some other properties they have, mostly involving clique graphs. Keywords: Chordal graph, dually chordal graph, clique tree, compatible tree. 1 Introduction 1.1 Definitions For a graph G, V (G) is the set of its vertices and E(G) is the set of its edges. A clique is a maximal set of pairwise adjacent vertices. The family of cliques of G is denoted by C(G). For v ∈ V (G), Cv will denote the family of cliques containing v. All graphs considered will be assumed to be connected. A set S ⊆ V (G) is a uv-separator if vertices u and v are in different connected components of G−S. It is minimal if no proper subset of S has the 1 Email: [email protected] aaURL: www.mate.unlp.edu.ar/~pdecaria 2 Email: [email protected] same property. S is a minimal vertex separator if there exist two nonadjacent vertices u and v such that S is a minimal uv-separator. S(G) will denote the family of all minimal vertex separators of G.