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. 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) 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.