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Minimally non-preperfect graphs of small maximum degree

Please always quote using this URN: urn:nbn:de:0297-zib-2971
  • 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}

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Metadaten
Author:Zsolt Tuza, Annegret Wagler
Document Type:ZIB-Report
Date of first Publication:1997/06/30
Series (Serial Number):ZIB-Report (SC-97-28)
ZIB-Reportnumber:SC-97-28
Published in:Appeared in: Graphs and Combinatorics 17 (2001) 759-773
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