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NP-hard graph problems and boundary classes of graphs

2007
*
Theoretical Computer Science
*

Any

doi:10.1016/j.tcs.2007.09.013
fatcat:aoqsgkkip5ck7ph7wwmg254uxe
*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. ...##
###
New results on maximum induced matchings in bipartite graphs and beyond

2013
*
Theoretical Computer Science
*

The maximum

doi:10.1016/j.tcs.2013.01.027
fatcat:fshlqajzyre4xhnfymbczo7g5u
*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] ). ...##
###
Critical properties of bipartite permutation graphs
[article]

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*. ...

##
###
Acyclic Matching in Some Subclasses of Graphs
[chapter]

2020
*
Lecture Notes in Computer Science
*

A

doi:10.1007/978-3-030-48966-3_31
fatcat:us2ztjou4vg2tatyxuonhlpfx4
*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*. ...##
###
Induced matchings in asteroidal triple-free graphs

2003
*
Discrete Applied Mathematics
*

An

doi:10.1016/s0166-218x(03)00390-1
fatcat:gtwpwwjlqbg5ldlcml4ofzitwq
*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*. ...##
###
Graph classes and the switch Markov chain for matchings

2015
*
Annales de la Faculté des Sciences de Toulouse
*

The total number

doi:10.5802/afst.1469
fatcat:wyu4owsg7rgavfb4o6zqakmynm
*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*. ...##
###
Statistical matching and subclassification with a continuous dose: characterization, algorithm, and application to a health outcomes study
[article]

2022
*
arXiv
*
pre-print

We discuss how to incorporate dose and use appropriate penalties to control the number

arXiv:2012.07182v2
fatcat:wkae66cnw5clpfmpnxlzacojpa
*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. ...##
###
Bi-induced sub graphs and stability number

2004
*
Yugoslav Journal of Operations Research
*

We define a 2-parametric hierarchy ( , ) m n CLAP

doi:10.2298/yjor0401027z
fatcat:yonlwfpmhrc4rohrvtc75igtyi
*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]. ...##
###
Clique separator decomposition of hole-free and diamond-free graphs and algorithmic consequences

2012
*
Discrete Applied Mathematics
*

This yields efficient algorithms for various optimization problems

doi:10.1016/j.dam.2011.10.031
fatcat:226muztrbfhyzgchj5i2q3y6xi
*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*...##
###
On greedy matching ordering and greedy matchable graphs
[chapter]

1997
*
Lecture Notes in Computer Science
*

We show that these

doi:10.1007/bfb0024498
fatcat:fuay63zsrfespkb4fqtohswmsy
*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*. ...##
###
An attractive class of bipartite graphs

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. ...

##
###
The Micro-world of Cographs
[chapter]

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. ...

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

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). ...

##
###
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

2008
*
Annals of Operations Research
*

However, the list-coloring problem is NP-complete for many

doi:10.1007/s10479-008-0391-5
fatcat:u46gfoggonefflcfocepfdwxsu
*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. ...
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