Dominating Cliques in Distance-Hereditary Graphs

1 I n t r o d u c t i o n A subset D C V is a dominating set in graph G = (V, E) iff for all vertices v E V \ D there is a vertex u E D with uv E E. It is a dominating clique in G iff D is a dominating set in G and a clique (i.e. for all u, v E D uv E E). There are many papers investigating the problem of finding minimum dominating sets in graphs with (and without) additional requirements to the dominating sets. The problems are in general NP-complete. For more special graphs the situation is sometimes better (for a bibliography on domination cf. [12], for a recent survey on special graph classes cf. [4]). Opposite to dominating set for a given ~ graph G a dominating clique does not necessarily exist. As was shown in [6] the problem whether a given graph has a dominating clique is NP-complete even for weakly chordal graphs (a graph G is weakly chordal iff G does not contain induced cycles C~ of length k > 5 and no complements ~ of such cycles). [14] and [15] investigate the dominating clique problem on strongly chordal and chordal graphs. A graph G is chordal iff it does not contain any induced cycle Ck of length k > 4. A graph G is strongly chordal iff it is chordal and each cycle in G of even length at least 6 has an odd chord (a chord xi, zj in a cycle C = (x i , . . . , x2k) of even length 2k is an odd chord if the distance in C between xi and zj is odd). Although for chordal graphs there is a simple criterion for the existence of dominating cliques the problem of finding a minimum (cardinality) dominating clique is NP-complete. In the ease of strongly chordal graphs the last problem is already polynomial-time solvable. *This work was partially supported by the VW-Stiftung Project No. 1/69041 e-mail address: [email protected]