A subset S of the vertices of a graph G is an outer-connected dominating set, if S is a dominating set of G and G − S is connected. The outer-connected domination number of G, denoted by γ̃c(G), is the minimum cardinality of an OCDS of G. In this paper we generalize the outer-connected domination in graphs. Many of the known results and bounds of outer-connected domination number are immediate c...

A sequence of vertices in a graph G is called a legal dominating sequence if every vertex in the sequence dominates at least one vertex not dominated by those vertices that precede it, and at the end all vertices of G are dominated. While the length of a shortest such sequence is the domination number of G, in this paper we investigate legal dominating sequences of maximum length, which we call...

Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular weak tail domination implies strong tail domination. In particular positive answer to Oleszkiewicz question would follow from the so-called Bernoulli conjecture. Introduction. This note is motivated by the following...

Using algebraic approach we implement a constant time algorithm for computing the domination numbers of the Cartesian products of paths and cycles. Closed formulas are given for domination numbers γ(Pn Ck) (for k ≤ 11, n ∈ N) and domination numbers γ(Cn Pk) and γ(Cn Ck) (for k ≤ 7, n ∈ N).

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 set S of vertices in a graph G = (V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. The total domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the tot...

We initiate the study of total outer-independent domination in graphs. A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \ D is independent. The total outer-independent domination number of a graph G is the minimum cardinality of a total outer-independent dominating set of G. First we discuss the ...

For a graph G, let f : V (G) → P({1, 2, . . . , k}) be a function. If for each vertex v ∈ V (G) such that f(v) = ∅ we have ∪u∈N(v)f(u) = {1, 2, . . . , k}, then f is called a k-rainbow dominating function (or simply kRDF) of G. The weight, w(f), of a kRDF f is defined as w(f) = ∑ v∈V (G) |f(v)|. The minimum weight of a kRDF of G is called the k-rainbow domination number of G, and is denoted by ...