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NP-hard graph problems and boundary classes of graphs

V.E. Alekseev, R. Boliac, D.V. Korobitsyn, V.V. Lozin
2007 Theoretical Computer Science  
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.  ...  Acknowledgement Vadim Lozin gratefully acknowledges the support of DIMAP -the Centre for Discrete Mathematics and its Applications at the University of Warwick.  ... 
doi:10.1016/j.tcs.2007.09.013 fatcat:aoqsgkkip5ck7ph7wwmg254uxe

New results on maximum induced matchings in bipartite graphs and beyond

Konrad K. Dabrowski, Marc Demange, Vadim V. Lozin
2013 Theoretical Computer Science  
The maximum induced matching problem is known to be APX-hard in the class of bipartite graphs.  ...  On the other hand, we show that for hypercubes (a proper subclass of regular bipartite graphs) the problem admits a simple solution.  ...  Algorithms to obtain such kernels have been found in some subclasses of (K s , K t,t )-free graphs (see e.g. [28] ).  ... 
doi:10.1016/j.tcs.2013.01.027 fatcat:fshlqajzyre4xhnfymbczo7g5u

Critical properties of bipartite permutation graphs [article]

Bogdan Alecu, Vadim Lozin, Dmitriy Malyshev
2020 arXiv   pre-print
In the present paper we identify critical subclasses of bipartite permutation graphs of various types.  ...  The class of bipartite permutation graphs enjoys many nice and important properties.  ...  a proper subclass of bipartite permutation graphs.  ... 
arXiv:2010.14467v1 fatcat:4limdlsygvdatfeyksncg3bm2u

Acyclic Matching in Some Subclasses of Graphs [chapter]

B. S. Panda, Juhi Chaudhary
2020 Lecture Notes in Computer Science  
A matching M in a graph G is called an acyclic matching if G[V (M )], the subgraph of G induced by the M -saturated vertices of G is acyclic.  ...  In this paper, we strengthen this result by showing that the decision version of the Acyclic Matching Problem remains NPcomplete for comb-convex bipartite graphs and dually-chordal graphs.  ...  is a subclass of bipartite graphs.  ... 
doi:10.1007/978-3-030-48966-3_31 fatcat:us2ztjou4vg2tatyxuonhlpfx4

Induced matchings in asteroidal triple-free graphs

Jou-Ming Chang
2003 Discrete Applied Mathematics  
An induced matching M of a graph G is a set of pairwise nonadjacent edges such that no two edges of M are joined by an edge in G.  ...  In this paper, we study the maximum induced matching problem on classes of graphs related to AT-free graphs.  ...  Note that the class of convex bipartite graphs properly contains bipartite permutation graphs as a subclass.  ... 
doi:10.1016/s0166-218x(03)00390-1 fatcat:gtwpwwjlqbg5ldlcml4ofzitwq

Graph classes and the switch Markov chain for matchings

Martin Dyer, Haiko Müller
2015 Annales de la Faculté des Sciences de Toulouse  
The total number of perfect matchings in a bipartite graph G is the permanent, which we denote by per(A) when A = A(G). Acknowledgements.  ...  To see that it is a subclass of chordal bipartite graphs, we permute rows so that β i ≤ β j when i < j. Now we can show that A(G) is -free. If not, there is a in some rows i, j and columns k , .  ...  These are usually called chain graphs in the graph theory literature [38] . They are a proper subclass of monotone graphs, which we consider in Section 2.4, and hence of chordal bipartite graphs.  ... 
doi:10.5802/afst.1469 fatcat:wyu4owsg7rgavfb4o6zqakmynm

Statistical matching and subclassification with a continuous dose: characterization, algorithm, and application to a health outcomes study [article]

Bo Zhang, Emily J. Mackay, Mike Baiocchi
2022 arXiv   pre-print
We discuss how to incorporate dose and use appropriate penalties to control the number of subclasses in the design.  ...  In this article, we propose two criteria for optimal subclassification based on subclass homogeneity in the context of having a continuous exposure dose, and propose an efficient polynomial-time algorithm  ...  In the context of non-bipartite matching with a continuous dose, we may also want to have some control over the size of subclasses and hence how many subclasses in a subclassification.  ... 
arXiv:2012.07182v2 fatcat:wkae66cnw5clpfmpnxlzacojpa

Bi-induced sub graphs and stability number

I.E. Zverovich, O.I. Zverovich
2004 Yugoslav Journal of Operations Research  
We define a 2-parametric hierarchy ( , ) m n CLAP of bi-hereditary classes of graphs, and show that a maximum stable set can be found in polynomial time within each class ( , ) m n CLAP .  ...  The classes can be recognized in polynomial time.  ...  In other words, a bi-induced subgraph F of a graph G is obtained from a bipartite induced sub graph F' of G by deleting some edges [possibly, none].  ... 
doi:10.2298/yjor0401027z fatcat:yonlwfpmhrc4rohrvtc75igtyi

Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences

Andreas Brandstädt, Vassilis Giakoumakis, Frédéric Maffray
2012 Discrete Applied Mathematics  
This yields efficient algorithms for various optimization problems in this class of graphs.  ...  We show that every (hole, paraglider)-free graph admits a clique separator decomposition into graphs of three very specific types.  ...  In a complete multipartite graph there is no induced matching of size 2, and in a matched co-bipartite graph there is no induced matching of size 3, so in either class the search for a maximum weight induced  ... 
doi:10.1016/j.dam.2011.10.031 fatcat:226muztrbfhyzgchj5i2q3y6xi

On greedy matching ordering and greedy matchable graphs [chapter]

Feodor F. Dragan
1997 Lecture Notes in Computer Science  
We show that these graphs, called greedy matchable graphs, are a subclass of weakly triangulated graphs and contain strongly chordal graphs and chordal bipartite graphs as proper subclasses.  ...  In this note a greedy algorithm is considered that computes a matching for a graph with a given ordering of its vertices, and those graphs are studied for which a vertex ordering exists such that the greedy  ...  W'e give a characterization of greedy matching orderings in terms of forbidden induced suborderings and present some properties of greedy matchable graphs.  ... 
doi:10.1007/bfb0024498 fatcat:fuay63zsrfespkb4fqtohswmsy

An attractive class of bipartite graphs

Rodica Boliac, Vadim Lozin
2001 Discussiones Mathematicae Graph Theory  
In this paper we propose a structural characterization for a class of bipartite graphs defined by two forbidden induced subgraphs.  ...  We show that the obtained characterization leads to polynomial-time algorithms for several problems that are NP-hard in general bipartite graphs.  ...  Note that all the listed classes are subclasses of chordal bipartite graphs, i.e., bipartite graphs without induced cycles of length more than 4.  ... 
doi:10.7151/dmgt.1151 fatcat:jfzexkpkc5capowbmnhmorgznu

The Micro-world of Cographs [chapter]

Bogdan Alecu, Vadim Lozin, Dominique de Werra
2020 Lecture Notes in Computer Science  
In the present paper, we identify several milestones in the world of cographs and create a hierarchy of graph parameters grounded on these milestones.  ...  Cographs constitute a small point in the atlas of graph classes. However, by zooming in on this point, we discover a complex world, where many parameters jump from finiteness to infinity.  ...  This is the class of (P 3 , K 3 )-free graphs, i.e., the class of bipartite graphs in U. S the class of stars, i.e., graphs of the form K 1,n and their induced subgraphs.  ... 
doi:10.1007/978-3-030-48966-3_3 fatcat:r6m7o5zetjbwvgbgfbew3mxbce

Induced matchings in strongly biconvex graphs and some algebraic applications [article]

Sara Saeedi Madani, Dariush Kiani
2019 arXiv   pre-print
In this paper, motivated by a question posed in AH, we introduce strongly biconvex graphs as a subclass of weakly chordal and bipartite graphs.  ...  We give a linear time algorithm to find an induced matching for such graphs and we prove that this algorithm indeed gives a maximum induced matching.  ...  Acknowledgments: The research of the first author was in part supported by a grant from IPM (No. 98130013). The research of the second author was in part supported by a grant from IPM (No. 98050212).  ... 
arXiv:1905.02640v1 fatcat:hh7gcctmgnac3cgzn2wsrk5cca

Page 4037 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews  
Summary: “In this paper, we present results on two subclasses of trapezoid graphs, including simple trapezoid graphs and trian- gle graphs (also known as PI graphs in [D. G. Corneil and P. A.  ...  Janse van Rensburg (3-YORK-MS; Toronto, ON) 2003f:05104 05C80 60C05 Kiwi, Marcos (RCH-UCS-EM; Santiago) ; Loebl, Martin (CZ-KARL-AM; Prague) Largest planar matching in random bipartite graphs.  ... 

Exploring the complexity boundary between coloring and list-coloring

Flavia Bonomo, Guillermo Durán, Javier Marenco
2008 Annals of Operations Research  
However, the list-coloring problem is NP-complete for many subclasses of perfect graphs.  ...  In this work we explore the complexity boundary between vertex coloring and list-coloring on such subclasses of perfect graphs where the former admits polynomial-time algorithms but the latter is NP-complete  ...  Acknowledgements We thank Dominique de Werra, Pavol Hell and Mario Valencia-Pabon for some interesting discussions about different topics on this work.  ... 
doi:10.1007/s10479-008-0391-5 fatcat:u46gfoggonefflcfocepfdwxsu
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