# Dominating cliques in graphs with hypertree structure

## 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.

- 1)
The dominating clique problem is generalized to r-domination (where each vertex has its “personal” domination radius).

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

- 4)
We investigate some related problems on chordal graphs and on dually chordal graphs.

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