Relationships between total domination, order, size, and maximum degree of graphs

Abstract

A total dominating set, S, in a graph, G, has the property that every vertex in G is adjacent to a vertex in S. The total dominating number, γt(G) of a graph G is the size of a minimum total dominating set in G. Let G be a graph with no component of size one or two and with Δ(G) ≥ 3. In 6, it was shown that |E(G)| ≤ Δ(G) (|V(G)|–γt(G)) and conjectured that |E(G)| ≤ (Δ(G)+3) (|V(G)|–γt(G))/2 holds. In this article, we prove that $\leq (\Delta(G)+ 2\sqrt{\Delta}) (|V(G)| - \gamma_{t}(G))/2$ holds and that the above conjecture is false as there for every Δ exist Δ‐regular bipartite graphs G with |E(G)| ≥ (Δ+0.1 ln(Δ)) (|V(G)|–γt(G))/2. The last result also disproves a conjecture on domination numbers from 8. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 325–337, 2007

Optimal linear‐Vizing relationships for (total) domination in graphs

“…The best-known lower and upper bounds (before this paper) to date on c Δ and r Δ for every ≥ Δ 3 are given by Rautenbach [18] and Yeo [24], and are summarized below.…”

“…As remarked by Yeo [24], "we do not have a guess if r Δ actually grows proportionally with ln(Δ) or Δ or some completely different function. "…”

More on limited packings in graphs

“…A domination set D of a graph G is called a total dominating set if G[D] has no isolated vertex, and the minimum cardinality of a total dominating set in G is called the total domination number of G, denoted by γ t (G). Total domination in graphs was introduced by Cockayne, Dawes, and Hedetniemi [6], and has been well studied (see, for example, [8,9,17,28]).…”

Equality in a linear Vizing-like relation that relates the size and total domination number of a graph

“…Sanchis [15] showed that if we restrict our attention to connected graphs with total domination number at least 5, then the bound in [4] can be improved slightly. The square dependence on n and γ t presented in [4,15] is improved in [9,16,19] into a linear dependence on n, γ t and ∆ by demanding a more even distribution of the edges by restricting the maximum degree ∆. In particular, the following linear Vizing-like relation relating the size of a graph and its order, total domination number, and maximum degree is established in [9,16].…”