Let D be a digraph. Its reverse digraph, D−1, is obtained by reversing all arcs of D. We show that the domination numbers of D and D−1 can be different if D is a Cayley digraph. The smallest groups admitting Cayley digraphs with this property are the alternating group A4 and the dihedral group D6, both on 12 elements. Then, for each n ≥ 6 we find a Cayley digraph D on the dihedral group Dn such...

Let G = (V,E) be a graph. A subset S of V is called a dominating set if each vertex of V −S has at least one neighbor in S. The domination number γ(G) equals the minimum cardinality of a dominating set in G. A minus dominating function on G is a function f : V → {−1, 0, 1} such that f(N [v]) = ∑ u∈N [v] f(u) ≥ 1 for each v ∈ V , where N [v] is the closed neighborhood of v. The minus domination ...

A function f de1ned on the vertices of a graph G = (V; E); f :V → {−1; 0; 1} is a minus dominating function if the sum of its values over any closed neighborhood is at least one. The weight of a minus dominating function is f(V ) = ∑ v∈V f(v). The minus domination number of a graph G, denoted by −(G), equals the minimum weight of a minus dominating function of G. In this paper, a sharp lower bo...

In a graph G, a vertex is said to dominate itself and its neighbors. The Domination game is a two player game played on a finite graph. Players alternate turns in choosing a vertex that dominates at least one new vertex. The game ends when no move is possible, that is when the set of chosen vertices forms a dominating set of the graph. One player (Dominator) aims to minimize the size of this se...

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 D be a finite and simple digraph with vertex set V (D) and arc set A(D). A signed Roman dominating function (SRDF) on the digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−[v] f(x) ≥ 1 for each v ∈ V (D), where N −[v] consists of v and all inner neighbors of v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The w...

If G = (V, E, σ) is a finite signed graph, a function f : V → {−1, 0, 1} is a minusdominating function (MDF) of G if f(u) +summation over all vertices v∈N(u) of σ(uv)f(v) ≥ 1 for all u ∈ V . In this paper we characterize signed paths and cycles admitting an MDF.

A function f defined on the vertices of a graph G = (V ,E), f : V → {−1, 0, 1} is a total minus dominating function (TMDF) if the sum of its values over any open neighborhood is at least one. The weight of a TMDF is the sum of its function values over all vertices. The total minus domination number, denoted by −t (G), of G is the minimum weight of a TMDF on G. In this paper, a sharp lower bound...