Bounds relating generalized domination parameters

Henning, A.M. and H.C. Swart, Bounds relating generalized domination parameters, Discrete Mathematics 120 (1993) 933105. The domination number r(G) and the total domination number y,(G) of a graph G are generalized to the K,-domination number 7x,(G) and the total K,-domination number y;_(G) for n>2, where y(G)=y,,(G) and Y~(G)=~~~(G), K,-connectivity is defined and, for every integer n >2, the existence of a Km-connected graph G of order at least n+ 1 for which yK,(G)+yi_(G)=((3n-2)/n’)p(G) is established. We conjecture that, if G is a K,-connected graph of order at least n+ 1, then g,“(G)+y:_(G)<((3n -2)/n’)p(G). This conjecture generalizes the result for n= 2 of Allan, Laskar and Hedetniemi. We prove the conjecture for n = 3. Further, it is shown that if G is a K,-connected graph of order at least 4 that satisfies the condition that, for each edge e of G, G-e contains at least one K,-isolated vertex, then y,,(G)+y’,,(G)<(3p)/4 and we show that this bound is best possible. The terminology and notation of [3] will be used throughout. In particular, G will denote a graph with vertex set V, edge set E, order p and size q. The neighbourhood N(u) of a vertex u is the set of all vertices that are adjacent with v, while the closed neighbourhood of v is the set N [v] = N(u) u {u}. If n is an integer, n 2 2 and u and v are distinct vertices of a graph G, then u and v are said to be &-adjacent vertices of G if there is a subgraph of G, isomorphic to K,, containing u and v. Therefore, u and u are K,-adjacent vertices of G if and only if u and u are adjacent vertices of G. A vertex that is contained in no subgraph of G, isomorphic to K,, is called a &-isolated vertex of G. Correspondence to: Henda C. Swart, Faculty Science Mathematics and Applied Mathematics, University of Natal, King George V Avenue, Durban 4001, South Africa. 0012-365X/93/$06.00