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On weakly diamond-free Berge graphs

Hacène Ait Haddadene, Sylvain Gravier
1996 Discrete Mathematics  
In this paper, we present a new class of graphs named weakly diamond-Jree (WDF) graphs and we prove for it the strong perfect graph conjecture, by exhibiting a polynomial sequential (,)-coloring algorithm  ...  This class contains chordal graphs and perfect line-graphs. * Corresponding author. 0012-365X/96/$15.00 ~ 1996 Elsevier Science B.V. All rights reserved PII S0012-365X(96)0003 1-3  ...  Here, we introduce a new class of graphs called weakly diamond-free (WDF) graphs, where diamond means K4 minus an edge.  ... 
doi:10.1016/0012-365x(96)00031-3 fatcat:ldutewpa75cd5njajgdixgngwu

Recognition Algorithm for Diamond-Free Graphs

Mihai Talmaciu, Elena Nechita
2007 Informatica  
diamond-free graphs which keeps the combinatorial structure of the graph by means of the decomposition, as well as an easy possibility to determine the clique number for the diamond-free graphs.  ...  In this paper we recall the notion of weakly decomposition, we recall some necessary and sufficient conditions for a graph to admit such a decomposition, we introduce the recognition algorithm for the  ...  The classes of graphs {diamond, co-diamond}free, {diamond, co-paw}-free, {diamond, 2K 2 }-free, {K 4 , co − diamond}-free have bounded clique-width. Theorem (see (Brandstadt et al., 2003) ).  ... 
doi:10.15388/informatica.2007.188 fatcat:vxb3nzatjfhfpl7nd2ac7oebg4

Critical and Anticritical Edges in Perfect Graphs [chapter]

Annegret Wagler
2001 Lecture Notes in Computer Science  
We ask in which perfect graphs critical and anticritical edges occur and how to find critical and anticritical edges in perfect graphs.  ...  We call an edge e of a perfect graph G critical if G − e is imperfect and say further that e is anticritical with respect to the complementary graph G.  ...  The graph depicted in Fig. 3(a) is critically and anticritically perfect and belongs to the classes of F -free Berge graphs where F is a claw, a diamond, or a K 4 .  ... 
doi:10.1007/3-540-45477-2_29 fatcat:iw2vgopldzeorpsy4g37jzwq5y

Counterexamples to three conjectures concerning perfect graphs

Stefan Hougardy
1993 Discrete Mathematics  
We will present counterexamples to a conjecture of Hoang on alternately orientable graphs, a conjecture of Hertz and de Werra on even pairs and to a conjecture of Reed on Berge graphs.  ...  All these three conjectures are related to perfect graphs. The Strong Perfect Graph Conjecture. A graph is perfect iff it is Berge.  ...  Acknowledgement I thank Chinh Ho&g for introducing me to the topic of perfect graphs and for many helpful comments.  ... 
doi:10.1016/0012-365x(93)90338-t fatcat:jm5qft66hfa2df2d4v3tcoay5q

Classes of perfect graphs

Stefan Hougardy
2006 Discrete Mathematics  
The Strong Perfect Graph Conjecture, suggested by Claude Berge in 1960, had a major impact on the development of graph theory over the last forty years.  ...  It has led to the definitions and study of many new classes of graphs for which the Strong Perfect Graph Conjecture has been verified.  ...  This class of graphs has been defined by Aït Haddadène and Maffray [1] who also proved the perfectness of these graphs. diamond-free Berge A graph is diamond-free Berge if it is a Berge graph that does  ... 
doi:10.1016/j.disc.2006.05.021 fatcat:gtgd6ojyr5gtdetickfby6n3xe

Page 2769 of Mathematical Reviews Vol. , Issue 97E [page]

1997 Mathematical Reviews  
diamond-free Berge graphs.  ...  Summary: “In this paper, we present a new class of graphs named weakly diamond-free graphs and we prove for it the strong perfect graph conjecture, by exhibiting a polynomial sequential w-coloring algorithm  ... 

Chair-Free Berge Graphs Are Perfect

Antonio Sassano
1997 Graphs and Combinatorics  
We prove that a Berge graph with no induced chair (chair-free) is perfect or, equivalently, that the Strong Perfect Graph Conjecture is true for chair-free graphs.  ...  The famous Berge's Strong Perfect Graph Conjecture asserts that every Berge graph is perfect. A chair is a graph with nodes {a, b, c, d, e} and edges {ab, bc, cd, eb}.  ...  On the other hand, the class of anti-chair-free graphs properly contains the class of diamond-free graphs and the class of paw-free graphs.  ... 
doi:10.1007/bf03353015 fatcat:ybs7c5inlnhdddztj5dck3javy

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

Andreas Brandstädt, Vassilis Giakoumakis, Frédéric Maffray
2012 Discrete Applied Mathematics  
We show that every (hole, paraglider)-free graph admits a clique separator decomposition into graphs of three very specific types.  ...  Clique separator decomposition, introduced by Whitesides and Tarjan, is one of the most important graph decompositions. A hole is a chordless cycle with at least five vertices.  ...  The classes of weakly chordal graphs and chordal bipartite graphs are also of importance here. A graph is weakly chordal (or weakly triangulated) if it is hole-free and antihole free.  ... 
doi:10.1016/j.dam.2011.10.031 fatcat:226muztrbfhyzgchj5i2q3y6xi

Graphs vertex-partitionable into strong cliques [article]

Ademir Hujdurović, Martin Milanič, Bernard Ries
2017 arXiv   pre-print
On the negative side, we prove NP-hardness of recognizing localizable graphs within the classes of weakly chordal graphs, complements of line graphs, and graphs of independence number three.  ...  Our results include a proof of the fact that every very well-covered graph is localizable and characterizations of localizable graphs within the classes of line graphs, triangle-free graphs, C_4-free graphs  ...  Acknowledgements The authors are grateful to Matjaž Krnc for helpful discussions, to EunJung Kim and Christophe Picouleau for feedback on an earlier draft, and to Rashid Zaare-Nahandi for providing them  ... 
arXiv:1609.06961v2 fatcat:nscvt2syzbcrhmuwg7oed6pvuu

Structure of classes of graphs defined by forbidding induced subgraphs [article]

Nicolas Trotignon
2013 arXiv   pre-print
It presents his work in structural graph theory, from 2001 to 2009.  ...  So, Theorem 2.30 gives in fact a much shorter proof of the perfectness of weakly triangulated graphs. Berge graphs behave in a way like weakly triangulated graphs.  ...  Theorem 2 . 20 ( 220 Harary and Holzmann [52] ) A graph G is the linegraph of a triangle-free graph if and only if G contains no claw and no diamond.  ... 
arXiv:1308.6678v1 fatcat:xgtoz7wi5nd7df3efa3lanoidq

Strong cliques in diamond-free graphs [article]

Nina Chiarelli, Berenice Martínez Barona, Martin Milanič, Jérôme Monnot, Peter Muršič
2020 arXiv   pre-print
On the positive side, we show that the following two problems whose computational complexity is open in general can be solved in linear time in the class of diamond-free graphs: Is every maximal clique  ...  We study strong cliques in the class of diamond-free graphs, from both structural and algorithmic points of view.  ...  Let G be a connected diamond-free graph. Then G is CIS if and only if G is either clique simplicial, G ∼ = K m,n for some m, n ≥ 2, or G ∼ = L(K n,n ) for some n ≥ 3.  ... 
arXiv:2006.13822v1 fatcat:54mubhy22be2nbabvqivr26cgm

Perfect graphs: a survey [article]

Nicolas Trotignon
2015 arXiv   pre-print
This is a survey about perfect graphs, mostly focused on the strong perfect graph theorem.  ...  Perfect graphs were defined by Claude Berge in the 1960s. They are important objects for graph theory, linear programming and combinatorial optimization.  ...  Theorem 12.9 (Chudnovsky and Seymour 2012) If G is a 3connected K 4 -free Berge graph with no even pair, and with no clique cutset, then one of G, G is the line graph of a bipartite graph.  ... 
arXiv:1301.5149v7 fatcat:sug4t6ty2jguxayq7a2mr4xnfm

The Strong Perfect Graph Conjecture: 40 years of attempts, and its resolution

F. Roussel, I. Rusu, H. Thuillier
2009 Discrete Mathematics  
The Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory.  ...  Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin, B.A.  ...  Since the diamond-free Berge graphs are perfect [104] *, a proof of this conjecture would imply the SPGT. Unfortunately, Conjecture 4.24 also is not true.  ... 
doi:10.1016/j.disc.2009.05.024 fatcat:kyvppq6dmndhlfn6j4jszqgvva

Combinatorial optimization with 2-joins

Nicolas Trotignon, Kristina Vušković
2012 Journal of combinatorial theory. Series B (Print)  
A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs.  ...  The techniques we develop are general enough to be easily applied to finding a maximum weighted stable set for another class of graphs known to be decomposable by 2-joins, namely the class of even-hole-free  ...  Originally, the project was to color Berge graphs without balanced skew partitions.  ... 
doi:10.1016/j.jctb.2011.06.002 fatcat:ed77rsm6ljh5pijxhfpl4ywfxa

On edge perfectness and classes of bipartite graphs

Haiko Müller
1996 Discrete Mathematics  
We define a notion of dependence for the edges of a graph and derive a concept of edge perfectness. We give some examples of classes of bipartite edge-perfect graphs.  ...  As mentioned above, this is a NP-completeness proof for ECC on general Ca-free graphs.  ...  Berge posed two conjectures according to this definition, the weaker one was proved by Lov~isz. Theorem 1 (Perfect graph theorem [22] ). A graph G is a-perfect if and only if G & ~o-perfect.  ... 
doi:10.1016/0012-365x(94)00350-r fatcat:mgf2oswmyjakjigvnhk45zf2ry
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