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    <id>297</id>
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    <language>eng</language>
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    <completedDate>1997-06-30</completedDate>
    <publishedDate>1997-06-30</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Minimally non-preperfect graphs of small maximum degree</title>
    <abstract language="eng">A graph $G$ is called preperfect if each induced subgraph $G' \subseteq G$ of order at least 2 has two vertices $x,y$ such that either all maximum cliques of $G'$ containing $x$ contain $y$, or all maximum indepentent sets of $G'$ containing $y$ contain $x$, too. Giving a partial answer to a problem of Hammer and Maffray [Combinatorica 13 (1993), 199-208], we describe new classes of minimally non-preperfect graphs, and prove the following characterizations: \begin{itemize} \item[(i)] A graph of maximum degree 4 is minimally non-preperfect if and only if it is an odd cycle of length at least 5, or the complement of a cycle of length 7, or the line graph of a 3-regular 3-connected bipartite graph. \item[(ii)] If a graph $G$ is not an odd cycle and has no isolated vertices, then its line graph is minimally non-preperfect if and only if $G$ is bipartite, 3-edge-connected, regular of degree $d$ for some $d \ge 3$, and contains no 3-edge-connected $d'$-regular subgraph for any $3 \le d' \le d$. \end{itemize}</abstract>
    <identifier type="serial">SC-97-28</identifier>
    <identifier type="opus3-id">298</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-2971</identifier>
    <enrichment key="SourceTitle">Appeared in: Graphs and Combinatorics 17 (2001) 759-773</enrichment>
    <author>Zsolt Tuza</author>
    <author>Annegret Wagler</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-97-28</number>
    </series>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
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    <file>https://opus4.kobv.de/opus4-zib/files/297/SC-97-28.pdf</file>
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