Abstract
In a recent series of articles R. Jamison and S. Olariu developed, starting from an extension of the notion of a cograph, a theory of the decomposition of graphs intoP 4-connected components. It turned out in their work that the algorithmic idea to exploit the unique tree structure of cographs can be generalized to graphs with simpleP 4-structure. In this paper we will show that deciding hamiltonicity and computing the path covering number are easy tasks forP 4-sparse andP 4-extendible graphs. We thereby generalize a result of H. A. Jung [8] concerning cographs.
Zusammenfassung
Vor kurzem haben R. Jamison und S. Olariu in einer Reihe von Arbeiten eine Theorie der Zerlegbarkeit von Graphen in ihreP 4-Zusammenhangskomponenten entwickelt und gezeigt, wie die Idee der Verwendung einer bis auf Isomorphie eindeutigen Baumstruktur zur Darstellung von Cographen und die damit verbundene Möglichkeit einer effektiven algorithmischen Behandlung dieser Graphen ausgedehnt werden kann auf Graphen, dieP 4-sparse oderP 4-extendible sind. Wir zeigen in dieser Arbeit, daß es für Graphen mit einer dieser beiden Eigenschaften leicht ist, die Weg-Überdeckungszahl zu berechnen und zu entscheiden, ob sie hamiltonsch sind oder nicht. Dabei verallgemeinern wir ein von H. A. Jung [8] gefundenes Resultat für Cographen.
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This work was done while the authors were visiting RUTCOR, Rutgers University.
Supported by SFB 303, DFG (German Research Association).
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Hochstättler, W., Tinhofer, G. Hamiltonicity in graphs with fewP 4's. Computing 54, 213–225 (1995). https://doi.org/10.1007/BF02253613
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DOI: https://doi.org/10.1007/BF02253613