Let G be a finite and simple graph with vertex set V (G), and let f : V (G) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑x∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ k for each x ∈ V (G), ...

Let G be a graph with the vertex set V (G) and edge set E(G). A function f : E(G) → {−1,+1} is said to be a signed star dominating function ofG if ∑ e∈EG(v) f(e) ≥ 1, for every v ∈ V (G), where EG(v) = {uv ∈ E(G) |u ∈ V (G)}. The minimum of the values of ∑ e∈E(G) f(e), taken over all signed star dominating functions f on G is called the signed star domination number of G and is denoted by γss(G...

A signed dominating function of a graph G with vertex set V is a function f : V → {−1, 1} such that for every vertex v in V the sum of the values of f at v and at every vertex u adjacent to v is at least 1. The weight of f is the sum of the values of f at every vertex of V . The signed domination number of G is the minimum weight of a signed dominating function of G. In this paper, we study the...

The domatic number d(G) of a graph G = (V,E) is the maximum order of a partition of V into dominating sets. Such a partition Π = {D1, D2, . . . , Dd} is called a minimal dominating d-partition if Π contains the maximum number of minimal dominating sets, where the maximum is taken over all d-partitions of G. The minimal dominating d-partition number Λ(G) is the number of minimal dominating sets ...

Let G be a finite and simple graph with vertex set V (G), and let f: V (G)→ {−1, 1} be a two-valued function. If k > 1 is an integer and ∑ x∈N[v] f(x) > k for each v ∈ V (G), where N [v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f1, f2, . . . , fd} of signed kdominating functions on G with the property that ∑ d i=1 fi(x) 6 k for each x ∈ V (G), is ca...

Let S be the set of minimal dominating sets of graph G and U, W ⊂ S with U ⋃ W = S and U ⋂ W = ∅. A Smarandachely mediate-(U,W ) dominating graph D m(G) of a graph G is a graph with V (D m(G)) = V ′ = V ⋃ U and two vertices u, v ∈ V ′ are adjacent if they are not adjacent in G or v = D is a minimal dominating set containing u. particularly, if U = S and W = ∅, i.e., a Smarandachely mediate-(S, ...

We consider the problem of incrementally computing a minimal dominating set of a directed graph after the insertion or deletion of a set of arcs. Earlier results have either focused on the study of the properties that minimum (not minimal) dominating sets preserved or lacked to investigate which update affects a minimal dominating set and in what ways. In this paper, we first show how to increm...