Total domination in generalized prisms and a new domination invariant

Abstract

In this paper we complement recent studies on the total domination of prisms by considering generalized prisms, i.e., Cartesian products of an arbitrary graph and a complete graph. By introducing a new domination invariant on a graph G, called the k-rainbow total domination number and denoted by γ krt (G), it is shown that the problem of finding the total domination number of a generalized prism G 2 K k is equivalent to an optimization problem of assigning subsets of {1, 2,. .. , k} to vertices of G. Various properties of the new domination invariant are presented, including, inter alia, that γ krt (G) = n for a nontrivial graph G of order n as soon as k ≥ 2∆(G). To prove the mentioned result as well as the closed formulas for the krainbow total domination number of paths and cycles for every k, a new weight-redistribution method is introduced, which serves as an efficient tool for establishing a lower bound for a domination invariant.

Bounding the k-rainbow total domination number

“…Proof. If a + b ≤ k the assertion is immediate (see Observation 4 in [20]). Now let a + b > k and let {A, B} be the bipartition of V (K a,b ) with A = {x 1 , x 2 , .…”

“…The main point of this section is to lower bound γ krt (G) in terms of γ(G). In [20] it was observed that for a graph G of order n where n > k > 1, it is always the case that max{k, γ(G)} ≤ γ krt (G). While the lower bound of k can be achieved, we will show that the lower bound of γ(G) cannot.…”

“…Since the seminal paper [4] introduced this invariant, it has been studied extensively, for example: its algorithmic properties [5,21], relevant graph operations and families [3,19], and general properties [16,18,22]. A k-rainbow total dominating function f of a graph G (a kRTDF for short) was introduced in [20] as a k-rainbow dominating function satisfying an additional condition that for every v ∈ V (G) such that f (v) = {i} for some i ∈ [k], there exists some u ∈ N (v) such that i ∈ f (u). The weight of a kRTDF is as in the case of a kRDF, ||f || = v∈V (G) |f (v)|.…”

“…For example, γ 2rt (C 4 ) = 4 since an optimal choice is to pick two vertices and assign each {1, 2}; another optimal choice simply assigns {1} to every vertex. The definition of the k-rainbow total domination number is motivated by wanting to understand total domination in the generalized prism; in [20] it was observed that for all k ≥ 1,…”