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Split graphs of Dilworth number 2

1985
*
Discrete Mathematics
*

Acknowledgment The authors express their gratitude to an anonymous referee who suggested the use

doi:10.1016/0012-365x(85)90040-8
fatcat:m7zekbpmuveflcq7u7fxt7m7o4
*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 ...##
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The Dilworth Number of Auto-Chordal-Bipartite Graphs
[article]

2013
*
arXiv
*
pre-print

We show that ACB

arXiv:1309.5787v1
fatcat:ywpa3immlvenlptcqfi3pxnvty
*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 ...##
###
The Dilworth Number of Auto-Chordal Bipartite Graphs

2014
*
Graphs and Combinatorics
*

We show that ACB

doi:10.1007/s00373-014-1471-8
fatcat:lnbouxvz5fc4db6gcub7x4562i
*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 ...##
###
On pairwise compatibility graphs having Dilworth number two

2014
*
Theoretical Computer Science
*

For

doi:10.1016/j.tcs.2013.12.015
fatcat:jb466rbv5fe53p3fml6fm2z3hy
*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. ...##
###
Split graphs with specified Dilworth numbers

1984
*
Canadian mathematical bulletin
*

Suppose that the

doi:10.4153/cmb-1984-006-6
fatcat:lqvd3p7xgvhedem6v6ag3yx3y4
*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 . ...##
###
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. ...##
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Pairwise Compatibility Graphs: A Survey

2016
*
SIAM Review
*

A

doi:10.1137/140978053
fatcat:b63dwsicgfa6ffuzauswrr3b3q
*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. ...##
###
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]

2014
*
Lecture Notes in Computer Science
*

The

doi:10.1007/978-3-319-04657-0_21
fatcat:m4c7o2e26jgmrorfi63l6hegae
*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] . ...##
###
Cographs: Eigenvalues and Dilworth Number
[article]

2018
*
arXiv
*
pre-print

The minimum

arXiv:1803.00246v2
fatcat:fsrxu5rowzdo7pvod6u7qq3mxq
*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). ...##
###
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

1990
*
Discrete Applied Mathematics
*

It can be shown that difference

doi:10.1016/0166-218x(90)90092-q
fatcat:s7ohur2kinc5xczid7g2hpt25y
*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 ...##
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A class of graphs with large rankwidth
[article]

2020
*
arXiv
*
pre-print

Korpelainen, Lozin, and Mayhill [

arXiv:2007.11513v2
fatcat:4tycpxtwbfbwxc4oiirva654jm
*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. ...##
###
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. ...##
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Graph classes with structured neighborhoods and algorithmic applications

2013
*
Theoretical Computer Science
*

Given a

doi:10.1016/j.tcs.2013.01.011
fatcat:n5fxjy7vq5andlu5mgk6sauvbi
*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. ...
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