Let k ≥ 1 be an integer, and let D = (V, A) be a finite and simple digraph in which dD(v) ≥ k for all v ∈ V . A function f : V −→ {−1, 1} is called a signed total k-dominating function (STkDF) if f(N−(v)) ≥ k for each vertex v ∈ V . The weight w(f) of f is defined by w(f) = ∑ v∈V f(v). The signed total k-domination number for a digraph D is γ kS(D) = min{w(f) | f is a STkDF of D}. In this paper...

4 Let γ(Pm2Pn) be the domination number of the Cartesian product of directed paths Pm and Pn for m,n ≥ 2. In [13] Liu and al. determined the value of γ(Pm2Pn) 6 for arbitrary n and m ≤ 6. In this work we give the exact value of γ(Pm2Pn) for any m,n and exhibit minimum dominating sets. 8 AMS Classification[2010]:05C69,05C38. 10

Let D be a finite and simple digraph with vertex set V (D). A signed total Roman k-dominating function (STRkDF) on D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−(v) f(x) ≥ k for each v ∈ V (D), where N−(v) consists of all vertices of D from which arcs go into v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The weight o...

Let G = (V,A) be a directed graph without parallel arcs, and let S ⊆ V be a set of vertices. Let the sequence S = S0 ⊆ S1 ⊆ S2 ⊆ · · · be defined as follows: S1 is obtained from S0 by adding all out-neighbors of vertices in S0. For k > 2, Sk is obtained from Sk−1 by adding all vertices w such that for some vertex v ∈ Sk−1, w is the unique out-neighbor of v in V \ Sk−1. We set M(S) = S0 ∪S1 ∪ · ...

The bondage number b(G) of a graph G is the smallest number of edges whose removal results in a graph with domination number greater than the domination number of G. Kang and Yuan [Bondage number of planar graphs. Discrete Math. 222 (2000), 191198] proved b(G) 6 min{8,∆+ 2} for every connected planar graph G, where ∆ is the maximum degree of G. Later Carlson and Develin [On the bondage number o...

We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

Motivated by the work on domination number of directed de Bruijn graphs and some its generalizations, in this paper we introduce a natural generalization (directed undirected), namely $t$-constrained graphs, where $t$ is positive integer, then study these graphs. Within definition Kautz correspond to 1-constrained 2-constrained respectively. This inherits many structural properties may have sim...

A three types competition model governed by directed last passage percolation on N 2 is considered. We prove that coexistence of the three types, i.e. the sets of vertices of the three types are simultaneously unbounded, occurs with positive probability. Moreover, the asymptotic angles formed by the two competition interfaces with the horizontal axis are determined and their probability of bein...

A total dominating set in a digraph G is a subset W of its vertices such that every vertex of G has an immediate successor in W . The total domination number of G is the size of the smallest total dominating set. We consider several lower bounds on the total domination number and conjecture that these bounds are strictly larger than g(G) − 1, where g(G) is the number of vertices of the smallest...