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A bound for the Dilworth number

C. van Nuffelen, M. van Wouwe
<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We give upper bounds for the Dilworth number of a graph.  ...  These bounds are formulated in terms of the rank of the adjacency matrix (vertex-vertex matrix) of the graph. 0012-365X/90/$3.50 0 1990, Elsevier Science Publishers B.V. (North-Holland)  ...  A chain is a subset C c V such that for any two vertices x and y of C, x my or y s x must hold. An antichain is a subset A c V such that for any x, y E A, x c y implies x = y.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(90)90153-9">doi:10.1016/0012-365x(90)90153-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/czzoj422zfbkxakagxanvab3p4">fatcat:czzoj422zfbkxakagxanvab3p4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170929132703/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/7b7/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMDEyMzY1eDkwOTAxNTM5.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f1/3d/f13df1038d6a5e105cf03b790b8a984ceeab6968.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(90)90153-9"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Cop number of graphs without long holes [article]

Vaidy Sivaraman
<span title="2019-12-30">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
As a consequence of our bound, we also give an inequality relating the cop number and the Dilworth number of a graph.  ...  We give a simple winning strategy for t-3 cops to capture a robber in the game of cops and robbers played in a graph that does not contain a hole of length at least t.  ...  The proof in this article was presented at the Barbados Graph Theory Workshop 2019 in Bellairs Research Institute in Holetown, Barbados, and the feedback from some of the participants was very useful in  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2001.00477v1">arXiv:2001.00477v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dbxssjqy7rgorkxbuy3x554cmu">fatcat:dbxssjqy7rgorkxbuy3x554cmu</a> </span>
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Page 331 of Annals of Mathematics Vol. 40, Issue 2 [page]

<span title="">1939</span> <i title="Princeton University Press Serials Division"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_annals-of-mathematics" style="color: black;">Annals of Mathematics </a> </i> &nbsp;
If the set of real numbers is bounded above, they still form a residuated structure provided that we take the product af equal to the least upper bound of the set whenever a + £ is greater than it.  ...  A function x on © to such a set of reals is called an evaluation of S if the following four conditions are satisfied: E 1. For every element a of S, ra is a uniquely determined real number.  ... 
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Cographs: Eigenvalues and Dilworth Number [article]

Ebrahim Ghorbani
<span title="2018-07-19">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ...  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.  ...  Acknowledgments The research of the author was in part supported by a grant from IPM (No. 95050114).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1803.00246v2">arXiv:1803.00246v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fsrxu5rowzdo7pvod6u7qq3mxq">fatcat:fsrxu5rowzdo7pvod6u7qq3mxq</a> </span>
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A greedy reduction algorithm for setup optimization

Ulrich Faigle, Rainer Schrader
<span title="">1992</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lx7dev2le5anbg6oarljwh7lie" style="color: black;">Discrete Applied Mathematics</a> </i> &nbsp;
Schrader, A greedy reduction algorithm for setup optimization, Discrete Applied Mathematics 35 (1992) 73-79. A reduction algorithm for setup optimization in general ordered sets is proposed.  ...  Moreover, the class of weakly cycle-free or.iers is introduced. All orders in this class are Dilworth optimal. Cycle-free orders and bipartite Dilworth optimal orders are proper subclasses.  ...  Hence the width w(P) (minus one) yields a lower bound for the setup number of P.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0166-218x(92)90297-n">doi:10.1016/0166-218x(92)90297-n</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qjjtbfgvxbd5fckyzsixmnashm">fatcat:qjjtbfgvxbd5fckyzsixmnashm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180723152303/https://ris.utwente.nl/ws/portalfiles/portal/6633697/Faigle92greedy.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/84/45/844561f95419b3eab11d119aba2d9a428b69ae54.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0166-218x(92)90297-n"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

A generalization of Witsenhausen's zero-error rate for directed graphs [article]

Gábor Simonyi, Ágnes Tóth
<span title="2014-06-03">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This scenario leads to the definition of a digraph parameter that generalizes Witsenhausen's zero-error rate for directed graphs.  ...  We investigate this new parameter for some specific directed graphs and explore its relations to other digraph parameters like Sperner capacity and dichromatic number.  ...  .) ♦ 3 Bounds on the Dilworth rate Relation to Sperner capacity and a lower bound Sperner capacity was introduced by Gargano, Körner and Vaccaro [16] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1406.0767v1">arXiv:1406.0767v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wlbzwjcwizas7dtou3ptwbyh4a">fatcat:wlbzwjcwizas7dtou3ptwbyh4a</a> </span>
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Page 3493 of Mathematical Reviews Vol. , Issue 98F [page]

<span title="">1998</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
This paper establishes a best possible upper bound for the Dil- worth number d(A) for rings of dimension at most one with fixed Hilbert function, and establishes the same best possible up- per bound for  ...  Eduard Bod'a (SK-KMSK;; Bratislava) 98f:13025 13H99 13E10 Sekiguchi, Hidemi |Ikeda, Hidemi] (J-NAGOC; Nagoya) The upper bound of the Dilworth number and the Rees number of Noetherian local rings with a  ... 
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Page 4120 of Mathematical Reviews Vol. , Issue 97G [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Our Dilworth number d(A) coincides with the usual Dilworth number of the partially ordered set P. Watanabe [op. cit.] proved that d(A) < r(A).  ...  Shi Zhong Pan (Dalian) 97g:13031 13D40 52B20 Herzog, Jiirgen (D-ESSN; Essen) ; Hibi, Takayuki (J-OSAKEGS; Toyonaka) Upper bounds for the number of facets of a simplicial complex.  ... 
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Dilworth Rate: A Generalization of Witsenhausen's Zero-Error Rate for Directed Graphs

Gabor Simonyi, Agnes Toth
<span title="">2015</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
This scenario leads to the definition of a digraph parameter that generalizes Witsenhausen's zero-error rate for directed graphs.  ...  We investigate this new parameter for some specific directed graphs and explore its relations to other digraph parameters like Sperner capacity and dichromatic number.  ...  Dichromatic number and upper bounds Now we show that the (logarithm of the) dichromatic number defined in [29] is an upper bound on the Dilworth rate.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tit.2014.2381668">doi:10.1109/tit.2014.2381668</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/h6dagkyibvhpxbuzy5b6afo4ka">fatcat:h6dagkyibvhpxbuzy5b6afo4ka</a> </span>
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A generalization of Witsenhausen's zero-error rate for directed graphs

Gabor Simonyi, Agnes Toth
<span title="">2014</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ux5tf45dtrgptnezraiii73msm" style="color: black;">2014 IEEE International Symposium on Information Theory</a> </i> &nbsp;
This scenario leads to the definition of a digraph parameter that generalizes Witsenhausen's zero-error rate for directed graphs.  ...  We investigate this new parameter for some specific directed graphs and explore its relations to other digraph parameters like Sperner capacity and dichromatic number.  ...  Dichromatic number and upper bounds Now we show that the (logarithm of the) dichromatic number defined in [29] is an upper bound on the Dilworth rate.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/isit.2014.6875357">doi:10.1109/isit.2014.6875357</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/isit/SimonyiT14.html">dblp:conf/isit/SimonyiT14</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7kd3l3mxznh3riffuv2vi5nvuq">fatcat:7kd3l3mxznh3riffuv2vi5nvuq</a> </span>
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The Dilworth Number of Auto-Chordal-Bipartite Graphs [article]

Anne Berry, Andreas Brandstädt, Konrad Engel
<span title="2013-09-23">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that ACB graphs have unbounded Dilworth number, and we characterize ACB graphs with Dilworth number k.  ...  The mirror (or bipartite complement) mir(B) of a bipartite graph B=(X,Y,E) has the same color classes X and Y as B, and two vertices x in X and y in Y are adjacent in mir(B) if and only if xy is not in  ...  We conclude this paper with the following open questions: 1. 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  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1309.5787v1">arXiv:1309.5787v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ywpa3immlvenlptcqfi3pxnvty">fatcat:ywpa3immlvenlptcqfi3pxnvty</a> </span>
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Page 2275 of Mathematical Reviews Vol. , Issue 85f [page]

<span title="">1985</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The aim of this paper is to give a characterization of split graphs with Dilworth number 3.  ...  Corresponding results for digraphs are given in the second article. Pavol Hell (Burnaby, B.C.) Nara, Chié 85f:05096 Split graphs with Dilworth number three. Natur. Sci. Rep.  ... 
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The Dilworth Number of Auto-Chordal Bipartite Graphs

Anne Berry, Andreas Brandstädt, Konrad Engel
<span title="2014-09-26">2014</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yxooi3wjmbgqtbp7l4evjeuj4i" style="color: black;">Graphs and Combinatorics</a> </i> &nbsp;
We show that ACB graphs have unbounded Dilworth number, and we characterize ACB graphs with Dilworth number k. *  ...  The mirror (or bipartite complement) mir(B) of a bipartite graph B = (X, Y, E) has the same color classes X and Y as B, and two vertices x ∈ X and y ∈ Y are adjacent in mir(B) if and only if xy / ∈ E.  ...  We conclude this paper with the following open questions: 1. 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  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00373-014-1471-8">doi:10.1007/s00373-014-1471-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lnbouxvz5fc4db6gcub7x4562i">fatcat:lnbouxvz5fc4db6gcub7x4562i</a> </span>
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Page 238 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 83, Issue 1 [page]

<span title="">1956</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_american-mathematical-society-transactions" style="color: black;">American Mathematical Society. Transactions of the American Mathematical Society </a> </i> &nbsp;
A description of a free lattice, FL(P), generated by any partially ordered set, P, has been given by R. P. Dilworth [2].  ...  In particular, the word problem is solved in these lattices, a canonical form is shown to exist for each word, and necessary and sufficient conditions are given for a finite subset of CF(P), considered  ... 
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Page 499 of Leisure Hour, an Illustrated Magazine for Home Reading Vol. 50, Issue [page]

<span title="">1901</span> <i title="Open Court Publishing Co"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_leisure-hour-an-illustrated-magazine-for-home-reading" style="color: black;">Leisure Hour, an Illustrated Magazine for Home Reading </a> </i> &nbsp;
But it made not a bit of difference; and we were still searching when Dilworth came rush- ing up the stairs, two at a bound, and dashed excitedly into the room. “There, what did I tell you?”  ...  ttatelindibinmsne eeiniece ae a We did. I never worked so hard before ; and I never before had an idea of the number and variety of the articles of furniture there were in the chambers.  ... 
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