A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted τ c (G), is the minimum cardinality of a clique-transversal set in G. In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound. Also, we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.
Let G = （V,A） be a digraph.A set T of vertices of G is a twin dominating set of G if for every vertex v ∈ V ／ T.There exist u,w ∈ T （possibly u = w） such that （u,v）,（v,w） ∈ A.The twin domination number γ＊（G） of G is the cardinality of a minimum twin dominating set of G.In this paper we consider the twin domination number in generalized Kautz digraphs GK（n,d）.In these digraphs,we establish bounds on the twin domination number and give a sufficient condition for the twin domination number attaining the lower bound.We give the exact values of the twin domination numbers by constructing minimum twin dominating sets for some special generalized Kautz digraphs.
The generalized Kautz digraphs have many good properties as interconnection network topologies. In this note, the bounds of the absorbant number for the generalized Kautz digraph are given, and some sufficient conditions for the absorbant number of the generalized Kautz digraph attaining the bounds are presented.
