Dominating cliques in graphs with hypertree structure
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.
The dominating clique problem is generalized to r-domination (where each vertex has its “personal” domination radius).
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  on chordal graphs is shown to be valid also in the more general case of r-domination.
We give a linear-time algorithm for the r-dominating clique problem on graphs with maximum neighbourhood orderings.
We investigate some related problems on chordal graphs and on dually chordal graphs.
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