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dc.contributorDepartment of Computingen_US
dc.contributor.advisorCao, Yixin (COMP)en_US
dc.contributor.advisorLi, Bo (COMP)en_US
dc.creatorWang, Shenghua-
dc.identifier.urihttps://theses.lib.polyu.edu.hk/handle/200/12986-
dc.languageEnglishen_US
dc.publisherHong Kong Polytechnic Universityen_US
dc.rightsAll rights reserveden_US
dc.titleT-perfect graphs and self-complementary graphsen_US
dcterms.abstractThe maximum-weight independent set problem is a fundamental NP-hard prob­lem. To gain a deeper understanding of its complexity, identifying graph classes where the problem can be solved in polynomial time has become a popular research area. Perfect graphs have emerged as one such class, characterized by their independent set polytope being fully described by trivial and clique inequalities. Inspired by the polyhedral characterization of perfect graphs, Chv´atal introduced t-perfect graphs, where the independent set polytope is fully described by trivial, edge, and odd-cycle inequalities. This pivotal characteristic enables the development of polynomial-time algorithms to solve the maximum-weight independent set problem specifically for t-perfect graphs. Given that t-perfect graphs are defined from a polyhedral perspective, a profound understanding of their structure is essential.en_US
dcterms.abstractWhile a full structural characterization of the class of t-perfect graphs is still at large, substantial advancements have been made for claw-free graphs [Bruhn and Stein, Math. Program. 2012] and P5-free graphs [Bruhn and Fuchs, SIAM J. Discrete Math. 2017]. We take one more step to characterize t-perfect graphs that are fork-free, and show that they are strongly t-perfect and three-colorable. We also present polynomial-time algorithms for recognizing and coloring these graphs.en_US
dcterms.abstractUnlike perfect graphs, t-perfect graphs are not closed under substitution or com­plementation. A full characterization of t-perfection with respect to substitution has been obtained by Benchetrit in his Ph.D. thesis. We attempt to understand t­-perfection with respect to complementation. In particular, we show that there are only five pairs of graphs such that both the graphs and their complements are mini­mally t-imperfect. We also identify all t-perfect graphs that are self-complementary.en_US
dcterms.abstractWe conduct a more in-depth study of self-complementary graphs. We study split graphs and pseudo-split graphs whose complements are isomorphic to themselves. These special subclasses of self-complementary graphs are actually the core of self-complementary graphs. Indeed, all realizations of forcibly self-complementary degree sequences are pseudo-split graphs. We also give formulas to calculate the number of self-complementary (pseudo-)split graphs of a given order, and show that Trotignon’s conjecture holds for all self-complementary split graphs.en_US
dcterms.extentx, 135 pages : color illustrationsen_US
dcterms.isPartOfPolyU Electronic Thesesen_US
dcterms.issued2024en_US
dcterms.educationalLevelPh.D.en_US
dcterms.educationalLevelAll Doctorateen_US
dcterms.LCSHPerfect graphsen_US
dcterms.LCSHGraph theoryen_US
dcterms.LCSHComputer science—Mathematicsen_US
dcterms.LCSHGraph algorithmsen_US
dcterms.LCSHHong Kong Polytechnic University -- Dissertationsen_US
dcterms.accessRightsopen accessen_US

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Please use this identifier to cite or link to this item: https://theses.lib.polyu.edu.hk/handle/200/12986