Graph Isomorphism Completeness for Chordal Bipartite Graphs and Strongly Chordal Graphs
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Estimating All Pairs Shortest Paths in Restricted Graph Families: a Unified Approach
Estimating All Pairs Shortest Paths in Restricted Graph Families: A Unified Approach (Extended Abstract) Feodor F. Dragan Dept. of Computer Science, Kent State University, Kent, Ohio 44242, USA [email protected] Abstract. In this paper we show that a very simple and efficient ap- proach can be used to solve the all pairs almost shortest path problem on the class of weakly chordal graphs and its different subclasses. Moreover, this approach works well also on graphs with small size of largest induced cycle and gives a unified way to solve the all pairs almost shortest path and all pairs shortest path problems on different graph classes includ- ing chordal, strongly chordal, chordal bipartite, and distance-hereditary graphs. 1 Introduction Let G =(V,E) be a finite, unweighted, undirected, connected and simple (i.e., without loops and multiple edges) graph. Let also |V | = n and |E| = m. The distance d(v, u) between vertices v and u is the smallest number of edges in a path connecting v and u. The all pairs shortest path problem is to compute d(v, u) for all pairs of vertices v and u in G. The all pairs shortest path (APSP) problem is one of the most fundamental graph problems. There has been a renewed interest in it recently (see [1,2,4,6,12], [15,29] for general (unweighted, undirected) graphs, and [3,5,7,10,11,14,20,21,24], [28,32,33] for special graph classes). For general graphs, the best result known is by Seidel [29], who showed that the APSP problem can be solved in O(M(n) log n) time where M(n) denotes the time complexity for matrix multiplication involving small integers only. -
Minimum Fill-In for Chordal Bipartite Graphs
Minimum fill-in for chordal bipartite graphs Citation for published version (APA): Kloks, A. J. J. (1993). Minimum fill-in for chordal bipartite graphs. (Universiteit Utrecht. UU-CS, Department of Computer Science; Vol. 9311). Utrecht University. Document status and date: Published: 01/01/1993 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
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. -
NP-Hard Graph Problems and Boundary Classes of Graphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 389 (2007) 219–236 www.elsevier.com/locate/tcs NP-hard graph problems and boundary classes of graphs V.E. Alekseeva, R. Boliacb, D.V. Korobitsyna, V.V. Lozinb,∗ a University of Nizhny Novgorod, Gagarina 23, Nizhny Novgorod, 603950, Russia b RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway NJ, 08854-8003, USA Received 1 September 2004; received in revised form 7 February 2007; accepted 10 September 2007 Communicated by A. Razborov Abstract Any graph problem, which is NP-hard in general graphs, becomes polynomial-time solvable when restricted to graphs in special classes. When does a difficult problem become easy? To answer this question we study the notion of boundary classes. In the present paper we define this notion in its most general form, describe several approaches to identify boundary classes and apply them to various graph problems. c 2007 Elsevier B.V. All rights reserved. Keywords: Computational complexity; Hereditary class of graphs 1. Introduction We study graph problems that are NP-hard in general, i.e. in the class of all graphs. When we restrict the class of input graphs, any of these problems can become tractable, i.e. solvable in polynomial time. When does a difficult problem become easy? Is there any boundary that separates classes of graphs where the problem remains NP-hard from those where it has a polynomial-time solution? To answer these questions, we study the notion of boundary classes of graphs. -
Cliques and Clubs⋆
Cliques and Clubs? Petr A. Golovach1, Pinar Heggernes1, Dieter Kratsch2, and Arash Rafiey1 1 Department of Informatics, University of Bergen, Norway, fpetr.golovach,pinar.heggernes,[email protected] 2 LITA, Universit´ede Lorraine - Metz, France, [email protected] Abstract. Clubs are generalizations of cliques. For a positive integer s, an s-club in a graph G is a set of vertices that induces a subgraph of G of diameter at most s. The importance and fame of cliques are evident, whereas clubs provide more realistic models for practical applications. Computing an s-club of maximum cardinality is an NP-hard problem for every fixed s ≥ 1, and this problem has attracted significant attention recently. We present new positive results for the problem on large and important graph classes. In particular we show that for input G and s, a maximum s-club in G can be computed in polynomial time when G is a chordal bipartite or a strongly chordal or a distance hereditary graph. On a superclass of these graphs, weakly chordal graphs, we obtain a polynomial-time algorithm when s is an odd integer, which is best possible as the problem is NP-hard on this clas for even values of s. We complement these results by proving the NP-hardness of the problem for every fixed s on 4-chordal graphs, a superclass of weakly chordal graphs. Finally, if G is an AT-free graph, we prove that the problem can be solved in polynomial time when s ≥ 2, which gives an interesting contrast to the fact that the problem is NP-hard for s = 1 on this graph class. -
New Min-Max Theorems for Weakly Chordal and Dually Chordal Graphs
New Min-Max Theorems for Weakly Chordal and Dually Chordal Graphs Arthur H. Busch1,, Feodor F. Dragan2, and R. Sritharan3, 1 Department of Mathematics, The University of Dayton, Dayton, OH 45469 [email protected] 2 Department of Computer Science, Kent State University, Kent, OH 44242 [email protected] 3 Department of Computer Science, The University of Dayton, Dayton, OH 45469 [email protected] Abstract. Adistance-k matching in a graph G is matching M in which the distance between any two edges of M is at least k.Adistance-2 matching is more commonly referred to as an induced matching. In this paper, we show that when G is weakly chordal, the size of the largest induced matching in G is equal to the minimum number of co-chordal subgraphs of G needed to cover the edges of G, and that the co-chordal subgraphs of a minimum cover can be found in polynomial time. Us- ing similar techniques, we show that the distance-k matching problem for k>1 is tractable for weakly chordal graphs when k is even, and is NP-hard when k is odd. For dually chordal graphs, we use properties of hypergraphs to show that the distance-k matching problem is solvable in polynomial time whenever k is odd, and NP-hard when k is even. Moti- vated by our use of hypergraphs, we define a class of hypergraphs which lies strictly in between the well studied classes of acyclic hypergraphs and normal hypergraphs. 1 Background and Motivation In this paper, all graphs are undirected, simple, and finite. -
The Maximum Induced Matching Problem for Some Subclasses of Weakly Chordal Graphs
THE MAXIMUM INDUCED MATCHING PROBLEM FOR SOME SUBCLASSES OF WEAKLY CHORDAL GRAPHS Thesis Submitted to The College of Arts and Science of the UNIVERSITY OF DAYTON in Partial Fulfillment of the Requirements for The Degree Master of Computer Science by Chandra Mohan Krishnamurthy UNIVERSITY OF DAYTON Dayton, Ohio December, 2009 THE MAXIMUM INDUCED MATCHING PROBLEM FOR SOME SUBCLASSES OF WEAKLY CHORDAL GRAPHS APPROVED BY: Barbara A. Smith, Ph.D. Dale Courte, Ph.D. Committee Member Committee Member Associate Professor, Chair and Associate Professor, Department of Computer Science Department of Computer Science R. Sritharan, Ph.D. Advisor Associate Professor, Department of Computer Science ii c Copyright by Chandra Mohan Krishnamurthy All rights reserved 2009 ABSTRACT THE MAXIMUM INDUCED MATCHING PROBLEM FOR SOME SUBCLASSES OF WEAKLY CHORDAL GRAPHS Name: Krishnamurthy, Chandra Mohan University of Dayton Advisor: Dr. R. Sritharan An induced matching in a graph is a set of edges such that no two edges in the set are joined by any third edge of the graph. An induced matching is maximum (MIM) if the number of edges in it is the largest among all possible induced matchings. It is known that finding the size of MIM in a graph is NP-hard even if the graph is bipartite. It is also known that the size of MIM in a chordal graph or in a weakly chordal graph can be computed in polynomial time. Specifically, the size of MIM can be computed in linear time for a chordal graph and in O(m3) time for a weakly chordal graph. This work demonstrates some algorithms for the maximum induced matching problem with complexity better than O(m3) for some subclasses of weakly chordal graphs. -
Treewidth of Chordal Bipartite Graphs
Treewidth of chordal bipartite graphs Citation for published version (APA): Kloks, A. J. J., & Kratsch, D. (1992). Treewidth of chordal bipartite graphs. (Universiteit Utrecht. UU-CS, Department of Computer Science; Vol. 9228). Utrecht University. Document status and date: Published: 01/01/1992 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. -
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]. -
A Min–Max Property of Chordal Bipartite Graphs with Applications
Graphs and Combinatorics (2010) 26:301–313 DOI 10.1007/s00373-010-0922-0 ORIGINAL PAPER A Min–Max Property of Chordal Bipartite Graphs with Applications Atif Abueida · Arthur H. Busch · R. Sritharan Received: 19 February 2008 / Revised: 2 February 2009 / Published online: 3 April 2010 © Springer 2010 Abstract We show that if G is a bipartite graph with no induced cycles on exactly 6 vertices, then the minimum number of chain subgraphs of G needed to cover E(G) equals the chromatic number of the complement of the square of line graph of G. Using this, we establish that for a chordal bipartite graph G, the minimum number of chain subgraphs of G needed to cover E(G) equals the size of a largest induced matching in G, and also that a minimum chain subgraph cover can be computed in polynomial time. The problems of computing a minimum chain cover and a largest induced matching are NP-hard for general bipartite graphs. Finally, we show that our results can be used to efficiently compute a minimum chain subgraph cover when the input is an interval bigraph. Keywords Chain cover · Induced matching · Bipartite graphs 1 Introduction and Motivation In this paper, all graphs G = (V, E) are simple and finite and consist of a set of vertices V and a set of edges E, which are unordered pairs of vertices. We follow the notation and terminology used in [22], and refer readers there for any unfamiliar terms. We use Ck to refer to a cycle on k vertices, n to refer to the number of vertices, and m to refer to the number of edges in a graph.