Contents

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Bounds on a Generalized Total Domination Parameter

Michael A. Henning1, Henda C. Swart2
1University of Zululand
2University of Natal.

Abstract

If n is an integer, n2, and u and v are vertices of a graph G, then u and v are said to be Kn-adjacent vertices of G if there is a subgraph of G, isomorphic to Kn, containing u and v. A total Kn-dominating set of G is a set D of vertices such that every vertex of G is Kn-adjacent to a vertex of D. The total Kn-domination number γKnt(G) of G is the minimum cardinality among the total Kn-dominating sets of vertices of G. It is shown that, for n{3,4,5}, if G is a graph with no Kn-isolated vertex, then γKnt(G)(2p)/n. Further, Kn-connectivity is defined and it is shown that, for n{3,4}, if G is a Kn-connected graph of order n+1, then γKnt(G)(2p)/(n+1). We establish that the upper bounds obtained are best possible.