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Dominating cliques in graphs with hypertree structure

  • Feodor F. Dragan
  • Andreas Brandstädt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 775)

Abstract

The use of (generalized) tree structure is one of the main topics in the field of efficient graph algorithms. The well-known partial k-tree approach belongs to this kind of research and bases on the tree structure of constant size-bounded maximal cliques. Without size bound on the cliques the tree structure remains helpful in some cases. We consider here graphs with this tree structure as well as a dual variant (in the sense of hypergraphs) of it which turns out to be useful for designing linear-time algorithms.

Elimination orderings of vertices are closely related to tree structures. Recently in several papers ([1], [4], [12], [21], [23]) graphs with maximum neighbourhood orderings were introduced and investigated. These orderings have some algorithmic consequences ([1], [9], [11]). Our aim in this paper is to carry over these techniques also to the dominating clique problem and to improve and generalize some recent results concerning this problem ([19], [20], [1]):
  1. 1)

    The dominating clique problem is generalized to r-domination (where each vertex has its “personal” domination radius).

     
  2. 2)

    We show that even for Helly graphs there exists a simple necessary and sufficient condition for the existence of dominating cliques, and the condition known from [20] on chordal graphs is shown to be valid also in the more general case of r-domination.

     
  3. 3)

    We give a linear-time algorithm for the r-dominating clique problem on graphs with maximum neighbourhood orderings.

     
  4. 4)

    We investigate some related problems on chordal graphs and on dually chordal graphs.

     

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Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Feodor F. Dragan
    • 1
  • Andreas Brandstädt
    • 2
  1. 1.Dept. of Math. and Cybern.Moldavian State UniversityChisinauMoldova
  2. 2.Universität -GH-Duisburg FB Math. FG Inf.DuisburgGermany

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