Finding Dominating Cliques Efficiently, in Strongly Chordal Graphs and Undirected Path Graphs
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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. -
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. -
Graph Isomorphism Completeness for Chordal Bipartite Graphs and Strongly Chordal Graphs
Discrete Applied Mathematics 145 (2005) 479–482 www.elsevier.com/locate/dam Notes Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs Ryuhei Ueharaa,1, Seinosuke Todab, Takayuki Nagoyac,∗ aNatural Science Faculty, Komazawa University, 1-23-1 Komazawa, Setagaya-Ku, Tokyo 154-8525, Japan bDepartment of Computer Science and System Analysis, College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-Ku, Tokyo 156-8550, Japan cDepartment of Mathematical Science, College of Sceince and Engineering, Tokyo Denki University, Hatoyama, Saitama 350-0394, Japan Received 6 November 2002; received in revised form 28 May 2004; accepted 19 June 2004 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, bipartite graphs, chordal graphs, comparability graphs, split graphs, and k-trees with unbounded k. On the other hand, the relative complexity of the GI problem for the above classes was unknown. We prove that deciding isomorphism of the classes are GI complete. © 2004 Elsevier B.V. All rights reserved. Keywords: 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 computational complexity theory three decades ago (see [8]). Although the problem is trivially in NP, the problem is not known to be in P and not known to be NP-complete either (see [9,11]). -
Complexity of Generalized Colourings of Chordal Graphs
COMPLEXITY OF GENERALIZED COLOURINGS OF CHORDAL GRAPHS by Juraj Stacho M.Sc., Comenius University, Slovakia, 2005 a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Computing Science c Juraj Stacho 2008 SIMON FRASER UNIVERSITY Spring 2008 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Juraj Stacho Degree: Doctor of Philosophy Title of thesis: Complexity of Generalized Colourings of Chordal Graphs Examining Committee: Dr. Joseph G. Peters Chair Dr. Pavol Hell, Senior Supervisor Dr. Ramesh Krishnamurti, Supervisor Dr. G´abor Tardos, SFU Examiner Dr. Ross M. McConnell, External Examiner, Computer Science Department, Colorado State University Date Approved: ii Abstract The generalized graph colouring problem (GCOL) for a fixed integer k, and fixed classes of graphs ,..., (usually describing some common graph properties), is to decide, for P1 Pk a given graph G, whether the vertex set of G can be partitioned into sets V1,...,Vk such that, for each i, the induced subgraph of G on V belongs to . It can be seen that GCOL i Pi generalizes many natural colouring and partitioning problems on graphs. In this thesis, we focus on generalized colouring problems in chordal graphs. The struc- ture of chordal graphs is known to allow solving many difficult combinatorial problems, such as the graph colouring, maximum clique and others, in polynomial, and in many cases in linear time. Our study of generalized colouring problems focuses on those problems in which the sets are characterized by a single forbidden induced subgraph. -
Relating Threshold Tolerance Graphs to Other Graph Classes
Relating threshold tolerance graphs to other graph classes Tiziana Calamoneri1? and Blerina Sinaimeri2 1 Sapienza University of Rome via Salaria 113, 00198 Roma, Italy. 2 INRIA and Universite´ de Lyon Universite´ Lyon 1, LBBE, CNRS UMR558, France. Abstract. A graph G = (V, E) is a threshold tolerance if it is possible to associate weights and tolerances with each node of G so that two nodes are adjacent exactly when the sum of their weights exceeds either one of their tolerances. Threshold tolerance graphs are a special case of the well-known class of tolerance graphs and generalize the class of threshold graphs which are also extensively studied in literature. In this note we relate the threshold tolerance graphs with other im- portant graph classes. In particular we show that threshold tolerance graphs are a proper subclass of co-strongly chordal graphs and strictly include the class of co-interval graphs. To this purpose, we exploit the relation with another graph class, min leaf power graphs (mLPGs). Keywords: threshold tolerance graphs, strongly chordal graphs, leaf power graphs, min leaf power graphs. 1 Introduction In the literature, there exist hundreds of graph classes (for an idea of the variety and the extent of this, see [2]), each one introduced for a different reason, so that some of them have been proven to be in fact the same class only in a second moment. It is the case of threshold graphs, that have been introduced many times with different names and different definitions (the interested reader can refer to [13]). Threshold graphs play an important role in graph theory and they model constraints in many combinatorial optimization problems [8, 12, 17]. -
Dually Chordal Graphs∗
SIAM J. DISCRETE MATH. c 1998 Society for Industrial and Applied Mathematics Vol. 11, No. 3, pp. 437{455, August 1998 007 DUALLY CHORDAL GRAPHS∗ ANDREAS BRANDSTADT¨ y , FEODOR DRAGANyz, VICTOR CHEPOIzx, AND VITALY VOLOSHINz Abstract. Recently in several papers, graphs with maximum neighborhood orderings were characterized and turned out to be algorithmically useful. This paper gives a unified framework for characterizations of those graphs in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal. These graphs are dual (in the sense of hypergraphs) to chordal graphs. By using the hypergraph approach in a systematical way new results are obtained, some of the old results are generalized, and some of the proofs are simplified. Key words. graphs, hypergraphs, tree structure, hypertrees, duality, chordal graphs, clique hypergraphs, neighborhood hypergraphs, disk hypergraphs, Helly property, chordality of line graphs, maximum neighborhood orderings, linear time recognition, doubly chordal graphs, strongly chordal graphs, bipartite incidence graphs AMS subject classifications. 05C65, 05C75, 68R10 PII. S0895480193253415 1. Introduction. The class of chordal graphs is a by now classical and well- understood graph class which is algorithmically useful and has several interesting characterizations. In the theory of relational database schemes there are close relation- ships between desirable properties of database schemes, acyclicity of corresponding hypergraphs, and chordality of graphs which corresponds to tree and Helly properties of hypergraphs [2], [5], [25]. Chordal graphs arise also in solving large sparse systems of linear equations [28], [36] and in facility location theory [13]. Recently a new class of graphs was introduced and characterized in [20], [6], [21], [39] which is defined by the existence of a maximum neighborhood ordering.