In this paper, we investigate domination number as well as signed domination numbers of Cay(G : S) for all cyclic group G of order n, where n in {p^m; pq} and S = { a^i : i in B(1; n)}. We also introduce some families of connected regular graphs gamma such that gamma_S(Gamma) in {2,3,4,5 }.

We briefly review known results about the signed edge domination number of graphs. In the case of bipartite graphs, the signed edge domination number can be viewed in terms of its bi-adjacency matrix. This motivates the introduction of the signed domination number of a (0, 1)-matrix. We investigate the signed domination number for various classes of (0, 1)-matrices, in particular for regular an...

A caterpillar is a tree with the property that after deleting all its vertices of degree 1 a simple path is obtained. The signed 2-domination number γ s (G) and the signed total 2-domination number γ st(G) of a graph G are variants of the signed domination number γs(G) and the signed total domination number γst(G). Their values for caterpillars are studied.

A {em weak signed Roman dominating function} (WSRDF) of a graph $G$ with vertex set $V(G)$ is defined as afunction $f:V(G)rightarrow{-1,1,2}$ having the property that $sum_{xin N[v]}f(x)ge 1$ for each $vin V(G)$, where $N[v]$ is theclosed neighborhood of $v$. The weight of a WSRDF is the sum of its function values over all vertices.The weak signed Roman domination number of $G...

For a graph G, a signed domination function of G is a two-colouring of the vertices of G with colours +1 and –1 such that the closed neighbourhood of every vertex contains more +1’s than –1’s. This concept is closely related to combinatorial discrepancy theory as shown by Füredi and Mubayi [J. Combin. Theory, Ser. B 76 (1999) 223–239]. The signed domination number of G is the minimum of the sum...

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...

Let k be a positive integer and G = (V,E) be a graph of minimum degree at least k − 1. A function f : V → {−1, 1} is called a signed k-dominating function of G if ∑ u∈NG[v] f(u) ≥ k for all v ∈ V . The signed k-domination number of G is the minimum value of ∑ v∈V f(v) taken over all signed k-dominating functions of G. The signed total k-dominating function and signed total k-domination number o...

In this paper we determine the exact values of the signed domination number, signed total domination number, and minus domination number of complete multipartite graphs, which substantially generalizes some previous results obtained for special subclasses of complete multipartite graphs such as cliques and complete bipartite graphs.

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) −→ {±1,±2, . . . ,±k} is said to be a signed star {k}-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. The signed star {k}-domination number of a graph G is γ{k}SS(G) = min{ ∑ e∈E f(e) | f is a S...