We show that the diameter of a total domination vertex-critical graph is at most 5(γt −1)/3, and that the diameter of an independent domination vertex-critical graph is at most 2(i− 1). For all values of γt ≡ 2 (mod 3) there exists a total domination vertex-critical graph with the maximum possible diameter. For all values of i ≥ 2 there exists an independent domination vertex-critical graph wit...

We find the maximum number of edges for a graph of given order and value of parameter for several domination parameters. In particular, we consider the total domination and independent domination numbers.

We prove a two-point concentration for the independent domination number of the random graph Gn,p provided p 2 ln(n) ≥ 64ln((lnn)/p).

for a graph $g$ let $gamma (g)$ be its domination number. we define a graph g to be (i) a hypo-efficient domination graph (or a hypo-$mathcal{ed}$ graph) if $g$ has no efficient dominating set (eds) but every graph formed by removing a single vertex from $g$ has at least one eds, and (ii) a hypo-unique domination graph (a hypo-$mathcal{ud}$ graph) if $g$ has at least two minimum dominating sets...

a set $s$ of vertices in a graph $g$ is a dominating set if every vertex of $v-s$ is adjacent to some vertex in $s$. the domination number $gamma(g)$ is the minimum cardinality of a dominating set in $g$. the annihilation number $a(g)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $g$ is at most the number of edges in $g$. in this p...

Domination parameters in random graphs G(n, p), where p is a fixed real number in (0, 1), are investigated. We show that with probability tending to 1 as n → ∞, the total and independent domination numbers concentrate on the domination number of G(n, p).

A Roman dominating function (RDF) on a graphG = (V,E) is a function f : V −→ {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF is the value f(V (G)) = ∑ u∈V (G) f(u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number γR(G)...

In this paper, we survey some new results in four areas of domination in graphs, namely: (1) the toughness and matching structure of graphs having domination number 3 and which are “critical” in the sense that if one adds any missing edge, the domination number falls to 2; (2) the matching structure of graphs having domination number 3 and which are “critical” in the sense that if one deletes a...

In this article we present characterizations of locally well-dominated graphs and locally independent well-dominated graphs, and a sufficient condition for a graph to be k-locally independent well-dominated. Using these results we show that the irredundance number, the domination number and the independent domination number can be computed in polynomial time within several classes of graphs, e....