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Split graphs of Dilworth number 2

C. Benzaken, P.L. Hammer, D. de Werra
1985 Discrete Mathematics  
Acknowledgment The authors express their gratitude to an anonymous referee who suggested the use of the above mentioned data structure in the algorithm.  ...  Split graphs and Dilworth number We first recall some definitions and results which are needed to characterize split graphs of Dilworth number at most 2.  ...  Introduction The purpose of this article is to give a simple proof that the class of split graphs with Dilworth number at most 2 coincides with the class of graphs G such that G and its complement G are  ... 
doi:10.1016/0012-365x(85)90040-8 fatcat:m7zekbpmuveflcq7u7fxt7m7o4

The Dilworth Number of Auto-Chordal-Bipartite Graphs [article]

Anne Berry, Andreas Brandstädt, Konrad Engel
2013 arXiv   pre-print
We show that ACB graphs have unbounded Dilworth number, and we characterize ACB graphs with Dilworth number k.  ...  We describe the relationship to some known graph classes such as interval and strongly chordal graphs and we present several characterizations of ACB graphs.  ...  Can one recognize ACB graphs in linear time? 2. Determine all (k, l)-critical ACB graphs G = (X, Y, E), i.e., all ACB graphs for which  ... 
arXiv:1309.5787v1 fatcat:ywpa3immlvenlptcqfi3pxnvty

The Dilworth Number of Auto-Chordal Bipartite Graphs

Anne Berry, Andreas Brandstädt, Konrad Engel
2014 Graphs and Combinatorics  
We show that ACB graphs have unbounded Dilworth number, and we characterize ACB graphs with Dilworth number k. *  ...  We describe the relationship to some known graph classes such as interval and strongly chordal graphs and we present several characterizations of ACB graphs.  ...  Can one recognize ACB graphs in linear time? 2. Determine all (k, l)-critical ACB graphs G = (X, Y, E), i.e., all ACB graphs for which  ... 
doi:10.1007/s00373-014-1471-8 fatcat:lnbouxvz5fc4db6gcub7x4562i

On pairwise compatibility graphs having Dilworth number two

T. Calamoneri, R. Petreschi
2014 Theoretical Computer Science  
For graphs with Dilworth number two that are also split graphs, i.e. split permutation graphs, we provide another way to compute T , w, d min and d max when these graphs are given in terms of their intersection  ...  The Dilworth number of a graph is the size of the largest subset of its nodes in which the close neighborhood of no node contains the neighborhood of another.  ...  The class of split permutation graphs is precisely the class of split graphs having Dilworth number at most two.  ... 
doi:10.1016/j.tcs.2013.12.015 fatcat:jb466rbv5fe53p3fml6fm2z3hy

Split graphs with specified Dilworth numbers

Chiê Nara, Iwao Sato
1984 Canadian mathematical bulletin  
Suppose that the Dilworth number of (I G , <) with respect to the vicinal preorder < is two and that of (K G , <) is an integer k.  ...  Let G be a split graph with the independent part I G and the complete part K G .  ...  Let G be an interval, split graph with Dilworth number at least two. Then V(G) is the largest number k such that G has an induced subgraph isomorphic to H k . x x .  ... 
doi:10.4153/cmb-1984-006-6 fatcat:lqvd3p7xgvhedem6v6ag3yx3y4

Page 2275 of Mathematical Reviews Vol. , Issue 85f [page]

1985 Mathematical Reviews  
The aim of this paper is to give a characterization of split graphs with Dilworth number 3.  ...  Nara, Chié 85f:05096 Split graphs with Dilworth number three. Natur. Sci. Rep. Ochanomizu Univ. 33 (1982), no. 1-2, 37-44.  ... 

Pairwise Compatibility Graphs: A Survey

Tiziana Calamoneri, Blerina Sinaimeri
2016 SIAM Review  
A graph G = (V, E) is a pairwise compatibility graph (PCG) if there exists an edgeweighted tree T and two nonnegative real numbers d min and dmax such that each leaf u of T is a node of V and there is  ...  In this article, we survey the state of the art concerning this class of graphs and some of its subclasses.  ...  w, c) are Dilworth t graphs; nevertheless, there are Dilworth t graphs, t ≥ 2, that do not belong to LP G ∩ mLP G.  ... 
doi:10.1137/140978053 fatcat:b63dwsicgfa6ffuzauswrr3b3q

Page 389 of Mathematical Reviews Vol. 57, Issue 2 [page]

1979 Mathematical Reviews  
Main theorems: A split graph which is simultaneously an interval and a comparability graph has Dilworth number | or 2 and conversely.  ...  The authors give several characterizations, in terms of Dilworth number and forbidden subgraphs, of those split graphs which are also interval or comparability graphs or both.  ... 

On Dilworth k Graphs and Their Pairwise Compatibility [chapter]

Tiziana Calamoneri, Rossella Petreschi
2014 Lecture Notes in Computer Science  
The Dilworth number of a graph is the size of the largest subset of its nodes in which the close neighborhood of no node contains the neighborhood of another one.  ...  Here we prove that graphs that are either LPGs or mLPGs of trees obtained connecting the centers of k stars with a path are Dilworth k graphs.  ...  Graphs with Dilworth number 1 and 2 have been deeply studied as they correspond to the classes of threshold and of threshold signed graphs, respectively [20] .  ... 
doi:10.1007/978-3-319-04657-0_21 fatcat:m4c7o2e26jgmrorfi63l6hegae

Cographs: Eigenvalues and Dilworth Number [article]

Ebrahim Ghorbani
2018 arXiv   pre-print
The minimum number of chains with respect to the vicinal preorder required to cover the vertex set of a graph G is called the Dilworth number of G.  ...  We prove that for any cograph G, the multiplicity of any eigenvalue λ0,-1, does not exceed the Dilworth number of G and show that this bound is tight. G. F. Royle [The rank of a cograph, Electron. J.  ...  Acknowledgments The research of the author was in part supported by a grant from IPM (No. 95050114).  ... 
arXiv:1803.00246v2 fatcat:fsrxu5rowzdo7pvod6u7qq3mxq

Page 630 of Mathematical Reviews Vol. , Issue 87b [page]

1987 Mathematical Reviews  
(CH-LSNP-P) Split graphs of Dilworth number 2. Discrete Math. 55 (1985), no. 2, 123-127. A graph G is split if its vertex set can be partitioned into a clique and an independent set.  ...  They also characterize the class of split graphs G with D(G) < 2 in terms of forbidden subgraphs. Finally they give a linear time algorithm for recognizing such graphs. J.  ... 

Difference graphs

Peter L. Hammer, Uri N. Peled, Xiaorong Sun
1990 Discrete Applied Mathematics  
It can be shown that difference graphs are bipartite. Properties of difference graphs are given in Section 2. It turns out that they are very similar to threshold graphs.  ...  Section 3 describes the polytope of bipartite degree sequences with a given bipartition, and shows that its extreme points are exactly the degree sequences of the difference graphs with the same bipartition  ...  The Dilworth number of a graph i with G is at most 2 if and only if there exist real numbers al, a2, . . . , a,, associate6 the vertices of G and positive real numbers S and T such that In Section 2, we  ... 
doi:10.1016/0166-218x(90)90092-q fatcat:s7ohur2kinc5xczid7g2hpt25y

A class of graphs with large rankwidth [article]

Chính Hoàng, Nicolas Trotignon
2020 arXiv   pre-print
Korpelainen, Lozin, and Mayhill [Split permutation graphs, Graphs and Combinatorics, 30(3):633–646, 2014] proved that there exist split graphs with Dilworth number 2 of arbitrarily large rankwidth, but  ...  We describe several graphs of arbitrarily large rankwidth (or equivalently of arbitrarily large cliquewidth).  ...  Split permutation graphs with Dilworth number 2 In [11] , it is proved that there exist split permutation graphs with Dilworth number 2 and arbitrarily large rankwidth.  ... 
arXiv:2007.11513v2 fatcat:4tycpxtwbfbwxc4oiirva654jm

Page 3809 of Mathematical Reviews Vol. , Issue 85i [page]

1985 Mathematical Reviews  
{For the entire collection see MR 84m:05004. } Nara, Chié; Sato, Iwao 85i1:05189 Split graphs with specified Dilworth numbers. Canad. Math. Bull. 27 (1984), no. 1, 43-47.  ...  Suppose that the Dilworth number of (Ig, <) with respect to the vicinal preorder < (defined in the paper) is two and that of (Kg, ~<) is an integer k.  ... 

Graph classes with structured neighborhoods and algorithmic applications

Rémy Belmonte, Martin Vatshelle
2013 Theoretical Computer Science  
Given a graph in any of the following graph classes: trapezoid graphs, circular permutation graphs, convex graphs, Dilworth k graphs, k-polygon graphs, circular arc graphs and complements of k-degenerate  ...  The boolean-width of a graph is related to the number of 1-neighbourhoods and our results imply that any of these graph classes have boolean-width O(log n).  ...  The Dilworth number of a graph is the largest number of pairwise incomparable vertices of the graph. A graph is a Dilworth k graph if it has Dilworth number k.  ... 
doi:10.1016/j.tcs.2013.01.011 fatcat:n5fxjy7vq5andlu5mgk6sauvbi
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