Characterization and Recognition of Some Opposition and Coalition
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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). -
Lecture 10: April 20, 2005 Perfect Graphs
Re-revised notes 4-22-2005 10pm CMSC 27400-1/37200-1 Combinatorics and Probability Spring 2005 Lecture 10: April 20, 2005 Instructor: L´aszl´oBabai Scribe: Raghav Kulkarni TA SCHEDULE: TA sessions are held in Ryerson-255, Monday, Tuesday and Thursday 5:30{6:30pm. INSTRUCTOR'S EMAIL: [email protected] TA's EMAIL: [email protected], [email protected] IMPORTANT: Take-home test Friday, April 29, due Monday, May 2, before class. Perfect Graphs k 1=k Shannon capacity of a graph G is: Θ(G) := limk (α(G )) : !1 Exercise 10.1 Show that α(G) χ(G): (G is the complement of G:) ≤ Exercise 10.2 Show that χ(G H) χ(G)χ(H): · ≤ Exercise 10.3 Show that Θ(G) χ(G): ≤ So, α(G) Θ(G) χ(G): ≤ ≤ Definition: G is perfect if for all induced sugraphs H of G, α(H) = χ(H); i. e., the chromatic number is equal to the clique number. Theorem 10.4 (Lov´asz) G is perfect iff G is perfect. (This was open under the name \weak perfect graph conjecture.") Corollary 10.5 If G is perfect then Θ(G) = α(G) = χ(G): Exercise 10.6 (a) Kn is perfect. (b) All bipartite graphs are perfect. Exercise 10.7 Prove: If G is bipartite then G is perfect. Do not use Lov´asz'sTheorem (Theorem 10.4). 1 Lecture 10: April 20, 2005 2 The smallest imperfect (not perfect) graph is C5 : α(C5) = 2; χ(C5) = 3: For k 2, C2k+1 imperfect. -
I?'!'-Comparability Graphs
Discrete Mathematics 74 (1989) 173-200 173 North-Holland I?‘!‘-COMPARABILITYGRAPHS C.T. HOANG Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, U.S.A. and B.A. REED Discrete Mathematics Group, Bell Communications Research, Morristown, NJ 07950, U.S.A. In 1981, Chv&tal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs. In this paper, we introduce a new class of perfectly orderable graphs, the &comparability graphs. This class generalizes comparability graphs in a natural way. We also prove a decomposition theorem which leads to a structural characteriza- tion of &comparability graphs. Using this characterization, we develop a polynomial-time recognition algorithm and polynomial-time algorithms for the clique and colouring problems for &comparability graphs. 1. Introduction A natural way to color the vertices of a graph is to put them in some order v*cv~<” l < v, and then assign colors in the following manner: (1) Consider the vertxes sequentially (in the sequence given by the order) (2) Assign to each Vi the smallest color (positive integer) not used on any neighbor vi of Vi with j c i. We shall call this the greedy coloring algorithm. An order < of a graph G is perfect if for each subgraph H of G, the greedy algorithm applied to H, gives an optimal coloring of H. An obstruction in a ordered graph (G, <) is a set of four vertices {a, b, c, d} with edges ab, bc, cd (and no other edge) and a c b, d CC. Clearly, if < is a perfect order on G, then (G, C) contains no obstruction (because the chromatic number of an obstruction is twu, but the greedy coloring algorithm uses three colors). -
(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. -
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. -
Box Representations of Embedded Graphs
Box representations of embedded graphs Louis Esperet CNRS, Laboratoire G-SCOP, Grenoble, France S´eminairede G´eom´etrieAlgorithmique et Combinatoire, Paris March 2017 Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. Ecological/food chain networks Sociological/political networks Fleet maintenance Boxicity d-box: the cartesian product of d intervals [x1; y1] ::: [xd ; yd ] of R × × Definition (Roberts 1969) The boxicity of a graph G, denoted by box(G), is the smallest d such that G is the intersection graph of some d-boxes. -
Improved Ramsey-Type Results in Comparability Graphs
Improved Ramsey-type results in comparability graphs D´anielKor´andi ∗ Istv´anTomon ∗ Abstract Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size n1=(r+1). This bound is known to be tight for r = 1. The question whether it is optimal for r ≥ 2 was studied by Dumitrescu and T´oth.We prove that it is essentially best possible for r = 2, as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size n1=3+o(1). Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G, nor its complement contains a complete bipartite graph with parts of size cn (log n)r . With this, we improve a result of Fox and Pach. 1 Introduction An old problem of Larman, Matouˇsek,Pach and T¨or}ocsik[11, 14] asks for the largest m = m(n) such that among any n convex sets in the plane, there are m that are pairwise disjoint or m that pairwise intersect. Considering the disjointness graph G, whose vertices correspond to the convex sets and edges correspond to disjoint pairs, this problem asks the Ramsey-type question of estimating the largest clique or independent set in G. -
P 4-Colorings and P 4-Bipartite Graphs Chinh T
P_4-Colorings and P_4-Bipartite Graphs Chinh T. Hoàng, van Bang Le To cite this version: Chinh T. Hoàng, van Bang Le. P_4-Colorings and P_4-Bipartite Graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2001, 4 (2), pp.109-122. hal-00958951 HAL Id: hal-00958951 https://hal.inria.fr/hal-00958951 Submitted on 13 Mar 2014 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. Discrete Mathematics and Theoretical Computer Science 4, 2001, 109–122 P4-Free Colorings and P4-Bipartite Graphs Ch´ınh T. Hoang` 1† and Van Bang Le2‡ 1Department of Physics and Computing, Wilfrid Laurier University, 75 University Ave. W., Waterloo, Ontario N2L 3C5, Canada 2Fachbereich Informatik, Universitat¨ Rostock, Albert-Einstein-Straße 21, D-18051 Rostock, Germany received May 19, 1999, revised November 25, 2000, accepted December 15, 2000. A vertex partition of a graph into disjoint subsets Vis is said to be a P4-free coloring if each color class Vi induces a subgraph without a chordless path on four vertices (denoted by P4). Examples of P4-free 2-colorable graphs (also called P4-bipartite graphs) include parity graphs and graphs with “few” P4s like P4-reducible and P4-sparse graphs. -
On Sources in Comparability Graphs, with Applications Stephan Olariu Old Dominion University
Old Dominion University ODU Digital Commons Computer Science Faculty Publications Computer Science 1992 On Sources in Comparability Graphs, With Applications Stephan Olariu Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/computerscience_fac_pubs Part of the Applied Mathematics Commons, and the Computer Sciences Commons Repository Citation Olariu, Stephan, "On Sources in Comparability Graphs, With Applications" (1992). Computer Science Faculty Publications. 128. https://digitalcommons.odu.edu/computerscience_fac_pubs/128 Original Publication Citation Olariu, S. (1992). On sources in comparability graphs, with applications. Discrete Mathematics, 110(1-3), 289-292. doi:10.1016/ 0012-365x(92)90721-q This Article is brought to you for free and open access by the Computer Science at ODU Digital Commons. It has been accepted for inclusion in Computer Science Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. Discrete Mathematics 110 (1992) 2X9-292 289 North-Holland Note On sources in comparability graphs, with applications S. Olariu Department of Computer Science, Old Dominion University, Norfolk, VA 235294162, USA Received 31 October 1989 Abstract Olariu, S., On sources in comparability graphs, with applications, Discrete Mathematics 110 (1992) 289-292. We characterize sources in comparability graphs and show that our result provides a unifying look at two recent results about interval graphs. An orientation 0 of a graph G is obtained by assigning unique directions to its edges. To simplify notation, we write xy E 0, whenever the edge xy receives the direction from x to y. A vertex w is called a source whenever VW E 0 for no vertex II in G. -
Combinatorial Optimization and Recognition of Graph Classes with Applications to Related Models
Combinatorial Optimization and Recognition of Graph Classes with Applications to Related Models Von der Fakult¨at fur¨ Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation vorgelegt von Diplom-Mathematiker George B. Mertzios aus Thessaloniki, Griechenland Berichter: Privat Dozent Dr. Walter Unger (Betreuer) Professor Dr. Berthold V¨ocking (Zweitbetreuer) Professor Dr. Dieter Rautenbach Tag der mundlichen¨ Prufung:¨ Montag, den 30. November 2009 Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.¨ Abstract This thesis mainly deals with the structure of some classes of perfect graphs that have been widely investigated, due to both their interesting structure and their numerous applications. By exploiting the structure of these graph classes, we provide solutions to some open problems on them (in both the affirmative and negative), along with some new representation models that enable the design of new efficient algorithms. In particular, we first investigate the classes of interval and proper interval graphs, and especially, path problems on them. These classes of graphs have been extensively studied and they find many applications in several fields and disciplines such as genetics, molecular biology, scheduling, VLSI design, archaeology, and psychology, among others. Although the Hamiltonian path problem is well known to be linearly solvable on interval graphs, the complexity status of the longest path problem, which is the most natural optimization version of the Hamiltonian path problem, was an open question. We present the first polynomial algorithm for this problem with running time O(n4). Furthermore, we introduce a matrix representation for both interval and proper interval graphs, called the Normal Interval Representation (NIR) and the Stair Normal Interval Representation (SNIR) matrix, respectively. -
The Micro-World of Cographs
The Micro-world of Cographs Bogdan Alecu Vadim Lozin Dominique de Werra June 10, 2020 B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 1 / 26 Any hereditary class can be described in terms of minimal forbidden induced subgraphs. Given a set S of graphs, Free(S) denotes the class of graphs with no induced subgraphs in S. A graph parameter is a function which associates to each graph a number. All parameters we consider are assumed to be hereditary, which means they do not increase when taking induced subgraphs. Examples: chromatic number, clique-width, ... Let p be a parameter and X a graph class. We say p is bounded in X if there is a constant k such that p(G) ≤ k for all G 2 X , and unbounded in X otherwise. We work with finite, simple, undirected graphs. A hereditary class (just \class" from now on) is a set of graphs closed under taking induced subgraphs. Basic definitions B. Alecu, V. Lozin, D. de Werra The Micro-world of Cographs June 10, 2020 2 / 26 Any hereditary class can be described in terms of minimal forbidden induced subgraphs. Given a set S of graphs, Free(S) denotes the class of graphs with no induced subgraphs in S. A graph parameter is a function which associates to each graph a number. All parameters we consider are assumed to be hereditary, which means they do not increase when taking induced subgraphs. Examples: chromatic number, clique-width, ... Let p be a parameter and X a graph class. -
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.