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Dominating the complements of bounded tolerance graphs and the complements of trapezoid graphs

J.Mark Keil, Patrice Belleville
<span title="">2004</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;
We investigate the complexity of several domination problems on the complements of bounded tolerance graphs and the complements of trapezoid graphs.  ...  We also prove that domination, connected domination and total domination are all NP-complete on co-trapezoid graphs. ?  ...  tolerance graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2003.04.004">doi:10.1016/j.dam.2003.04.004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/y4ru4tjkore7rp5m5tuf467bdm">fatcat:y4ru4tjkore7rp5m5tuf467bdm</a> </span>
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The Recognition of Tolerance and Bounded Tolerance Graphs [article]

George B. Mertzios, Ignasi Sau, Shmuel Zaks
<span title="2010-02-03">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
For our reduction we extend the notion of an acyclic orientation of permutation and trapezoid graphs.  ...  be efficiently recognizable as they are a natural special case of trapezoid graphs (which can be recognized in polynomial time) and share a very similar structure with them.  ...  The formula φ is NAE-satisfiable if and only if the trapezoid graph H φ is a bounded tolerance graph.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1001.3251v2">arXiv:1001.3251v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tgxyfqlvhzcovnzk4cj2nxz7gm">fatcat:tgxyfqlvhzcovnzk4cj2nxz7gm</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1001.3251/1001.3251.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> File Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/05/88/058886aaad229a7ec4fac494d8beb9a5d08ab009.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1001.3251v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

The Recognition of Tolerance and Bounded Tolerance Graphs

George B. Mertzios, Ignasi Sau, Shmuel Zaks
<span title="">2011</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7dys7zoberdktmxyjciuy5bnse" style="color: black;">SIAM journal on computing (Print)</a> </i> &nbsp;
The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes  ...  Theorem 3 . 2 . 32 The formula φ is NAE-satisfiable if and only if the trapezoid graph H φ is a bounded tolerance graph.  ...  tolerance and bounded tolerance graphs are in the class NP [15] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/090780328">doi:10.1137/090780328</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oiqppw7hifespi7c4laqsewitu">fatcat:oiqppw7hifespi7c4laqsewitu</a> </span>
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On the intersection of tolerance and cocomparability graphs

George B. Mertzios, Shmuel Zaks
<span title="">2016</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;
Furthermore, it is known that the intersection of tolerance and cocomparability graphs is contained in the class of trapezoid graphs.  ...  It has been conjectured by Golumbic and Monma in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs.  ...  We first review in Section 2 some properties of tolerance and trapezoid graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2014.10.025">doi:10.1016/j.dam.2014.10.025</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gxenkzpnybbjxfhxpzwvhfbhby">fatcat:gxenkzpnybbjxfhxpzwvhfbhby</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170706015923/http://community.dur.ac.uk/george.mertzios/papers/Conf/Conf_Intersection-Conjecture.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/68/4c/684c72c5d3aea3126de0b00382687bbdd7079651.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.dam.2014.10.025"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> elsevier.com </button> </a>

On the Intersection of Tolerance and Cocomparability Graphs [chapter]

George B. Mertzios, Shmuel Zaks
<span title="">2010</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Furthermore, it is known that the intersection of tolerance and cocomparability graphs is contained in the class of trapezoid graphs.  ...  It has been conjectured by Golumbic and Monma in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs.  ...  We first review in Section 2 some properties of tolerance and trapezoid graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-17517-6_22">doi:10.1007/978-3-642-17517-6_22</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oe2oamcbivgjddrz6dmavxmklm">fatcat:oe2oamcbivgjddrz6dmavxmklm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170706015923/http://community.dur.ac.uk/george.mertzios/papers/Conf/Conf_Intersection-Conjecture.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/68/4c/684c72c5d3aea3126de0b00382687bbdd7079651.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-17517-6_22"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Computing Terminal Reliability of Multi-Tolerance Graphs

Chien-Min CHEN, Min-Sheng LIN
<span title="">2016</span> <i title="Institute of Electronics, Information and Communications Engineers (IEICE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/xosmgvetnbf4zpplikelekmdqe" style="color: black;">IEICE transactions on information and systems</a> </i> &nbsp;
multi-tolerance graphs.  ...  This work demonstrates that on multi-tolerance graphs, the 2-terminal reliability problem can be solved in polynomial-time and the results can be extended to the K-terminal reliability problem on bounded  ...  Acknowledgments The authors would like to thank the Ministry of Science and Technology of the Republic of China, Taiwan, for financially supporting this research under Contract No.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1587/transinf.2015edp7418">doi:10.1587/transinf.2015edp7418</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2hropszvyra5lpep242kwudzj4">fatcat:2hropszvyra5lpep242kwudzj4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20181102175007/https://www.jstage.jst.go.jp/article/transinf/E99.D/7/E99.D_2015EDP7418/_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/1e/fa/1efae2adf6e8621a07458509c32f2a994f2fbab4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1587/transinf.2015edp7418"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Triangulating multitolerance graphs

Andreas Parra
<span title="">1998</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;
In this paper we introduce a new class of graphs which generalize both the tolerance and the trapezoid graphs, the multitolerance graphs.  ...  These results complement the recent results of Habib and Miihring [ 18, 251 about the treewidth and pathwidth of cocomparability graphs and graphs without asteroidal triples, and those of Kloks et al.  ...  MGhring, and Petra Scheffler for valuable discussions, and for helpful comments that improved the manuscript.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0166-218x(98)00026-2">doi:10.1016/s0166-218x(98)00026-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zcw6dywk7vc5viku7tgkdjjfyy">fatcat:zcw6dywk7vc5viku7tgkdjjfyy</a> </span>
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On Pairwise Compatibility of Some Graph (Super)Classes [article]

Tiziana Calamoneri, Blerina Sinaimeri, Mattia Gastaldello
<span title="2015-04-24">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
tolerance graphs.  ...  The non-inclusion of some superclasses (trapezoid, permutation and rectangle intersection graphs) follows.  ...  Permutation and trapezoid graphs, superclasses of circular arc graphs, are trivially not in PCG, too.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1504.06454v1">arXiv:1504.06454v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xje4ry55arbkffejgypw6elvb4">fatcat:xje4ry55arbkffejgypw6elvb4</a> </span>
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Domination graphs: examples and counterexamples

Irena Rusu, Jeremy Spinrad
<span title="">2001</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;
We give a counterexample to a conjecture of Dahlhaus et al. [3] claiming that the Special Quadratic Consensus Method yields a polynomial-time recognition for domination graphs, and discuss several new  ...  properties of domination graphs. ?  ...  We also show that domination graphs contain the (incomparable) classes of trapezoid graphs and tolerance graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0166-218x(00)00274-2">doi:10.1016/s0166-218x(00)00274-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cpoyy72fqjf3pi63pnggqz5udm">fatcat:cpoyy72fqjf3pi63pnggqz5udm</a> </span>
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Tolerance Graphs [chapter]

Martin Golumbic
<span title="2013-11-26">2013</span> <i title="Chapman and Hall/CRC"> Discrete Mathematics and Its Applications </i> &nbsp;
, 25 of recognizing permutation graphs, 163 of recognizing tolerance graphs, 166 of recognizing transitive orien- tations, 24 of recognizing trapezoid graphs and orders, 153, 163 of recognizing  ...  order, 112, 178 trampoline, 38 transitive orientation, 22, 50, 137 transitively orientable, 22, 50 trapezoid graph, 33, 57, 68, 81, 117, 119, 166 trapezoid graph recognition, 153 trapezoid order  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1201/b16132-66">doi:10.1201/b16132-66</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j2i3oyr2e5au7mtjsdmhmcaezm">fatcat:j2i3oyr2e5au7mtjsdmhmcaezm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20030509141129/http://www.cs.haifa.ac.il:80/~golumbic/tol-cont-openq-bib.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/4b/64/4b64ff7561ddbd2f74cfe246748b3427e5b5c278.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1201/b16132-66"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Tolerance graphs and orders [chapter]

Stefan Felsner
<span title="">1993</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
As consequences we are able to give obstructions for the class of bounded tolerance graphs and to give an example of a graph which is alternatingly orientable but not a tolerance graph.  ...  We show that if a tolerance graph is the complement of a comparability graph it is a trapezoid graph, i.e., the complement of an order of interval dimension at most 2.  ...  Therefore if T is a tolerance graph, then T is a bounded tolerance graph. If T is a bounded tolerance graph, then trivially it is a tolerance graph, and by Theorem 3 it is a trapezoid graph.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-56402-0_32">doi:10.1007/3-540-56402-0_32</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7f7x2wat45ccxjpn7fffkf4zwi">fatcat:7f7x2wat45ccxjpn7fffkf4zwi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20080424161109/http://www.math.tu-berlin.de/~felsner/Paper/tolgr.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/81/bb/81bb52802a2effabf3552ad216a39fee4294b948.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-56402-0_32"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Tolerance graphs, and orders

Stefan Felsner
<span title="">1998</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukzsjb6a6zhyxnjl6nb2mmjc6m" style="color: black;">Journal of Graph Theory</a> </i> &nbsp;
As consequences we are able to give obstructions for the class of bounded tolerance graphs and to give an example of a graph which is alternatingly orientable but not a tolerance graph.  ...  We show that if a tolerance graph is the complement of a comparability graph it is a trapezoid graph, i.e., the complement of an order of interval dimension at most 2.  ...  Therefore if T is a tolerance graph, then T is a bounded tolerance graph. If T is a bounded tolerance graph, then trivially it is a tolerance graph, and by Theorem 3 it is a trapezoid graph.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/(sici)1097-0118(199807)28:3&lt;129::aid-jgt2&gt;3.0.co;2-m">doi:10.1002/(sici)1097-0118(199807)28:3&lt;129::aid-jgt2&gt;3.0.co;2-m</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j2forkw2tvaavckd3pqezjjmru">fatcat:j2forkw2tvaavckd3pqezjjmru</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20080424161109/http://www.math.tu-berlin.de/~felsner/Paper/tolgr.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/81/bb/81bb52802a2effabf3552ad216a39fee4294b948.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/(sici)1097-0118(199807)28:3&lt;129::aid-jgt2&gt;3.0.co;2-m"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> wiley.com </button> </a>

Trapezoid graphs and their coloring

Ido Dagan, Martin Charles Golumbic, Ron Yair Pinter
<span title="">1988</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;
We provide an optimal coloring algorithm for trapezoid graphs that runs in time o(nk), where n is the number of nodes and k is the chromatic number of the graph.  ...  'vile show that this new class properly contains the union of the two former classes, and that trapezoid graphs are equivalent to the incomparability graphs of partially ordered sets having interval order  ...  The classes of proper tolerance graphs and trapezoid graphs share a number of common properties: They include all interval graphs and all permutation graphs, they are themseives coTR0 graphs, and they  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0166-218x(88)90032-7">doi:10.1016/0166-218x(88)90032-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tvjlz6ycnvdhtnrtemn7ledwfu">fatcat:tvjlz6ycnvdhtnrtemn7ledwfu</a> </span>
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An Intersection Model for Multitolerance Graphs: Efficient Algorithms and Hierarchy [chapter]

George B. Mertzios
<span title="2011-01-23">2011</span> <i title="Society for Industrial and Applied Mathematics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ja7dpjgz2nhjlovlil3ij5m7ei" style="color: black;">Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms</a> </i> &nbsp;
The author would like to thank Sagi Snir from the University of Haifa, Israel, for indicating the application of multitolerance graphs in the comparison of DNA sequences.  ...  Therefore, the graph T 2 is a multitolerance but not a trapezoid graph, while the graph B is a multitolerance but not a tolerance graph, and thus multitolerance graphs strictly include both tolerance and  ...  Theorem 7 Multitolerance graphs strictly include tolerance and trapezoid graphs, while they are strictly included in weakly chordal graphs. Proof.  ... 
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An Intersection Model for Multitolerance Graphs: Efficient Algorithms and Hierarchy

George B. Mertzios
<span title="2013-02-07">2013</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qhi3z76be5c5xeihac5cyiid3m" style="color: black;">Algorithmica</a> </i> &nbsp;
The author would like to thank Sagi Snir from the University of Haifa, Israel, for indicating the application of multitolerance graphs in the comparison of DNA sequences.  ...  Therefore, the graph T 2 is a multitolerance but not a trapezoid graph, while the graph B is a multitolerance but not a tolerance graph, and thus multitolerance graphs strictly include both tolerance and  ...  Theorem 7 Multitolerance graphs strictly include tolerance and trapezoid graphs, while they are strictly included in weakly chordal graphs. Proof.  ... 
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