On a-total domination in prisms and mobius ladders

Abstract

Let G = (V, E) be a graph with no isolated vertex. A subset of vertices S is a total dominating set if every vertex of G is adjacent to some vertex of S. For some q with 0 < a < 1, a total dominating set S in G is an a-total dominating set if for every vertex v € V \ S, |AT(t,)nS| a\N(v)\. The a-total domination number of G, denoted by 7fti(£), is the minimum cardinality of an a-total dominating set of G. In [1], Henning and Rad posed the following question: Let G be a connected cubic graph with order n. Is it true that at(G) < 23" for \ < a < I and 7nt{G) < 34n for \ < a < 1 In this paper, we find a-total domination numbers for two classes of connected cubic graphs, namely the prisms and the Nlobius ladders. All the exactly values are less than the bounds in the question. We give apositive answer toward this question on these two classes of connected cubic graphs. © 2018 Utilitas Mathematica Publishing Incorporated. All rights reserved.