Let G = (V , E) be a simple graph on vertex set V and define a function f : V → {−1,1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. We give a sharp lower bound on the signed do...

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, 1} 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)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that ∑d i=1 fi(e) ...

For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (G) with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2, . . . , k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set {f1, f2, ....

In this paper we deal with the set of k-additive belief functions dominating a given capacity. We follow the line introduced by Chateauneuf and Jaffray for dominating probabilities and continued by Grabisch for general k-additive measures. First, we show that the conditions for the general k-additive case lead to a very wide class of functions and this makes that the properties obtained for pro...

The generic-level taxa included in Pterostichini and Loxandrini from New Caledonia are reviewed and a key to genera and species provided. Two new genera are described, Paniestichus and Abacophrastus, with the following new species: Paniestichus subsolianus, Abacophrastus millei, Abacophrastus chapes, Abacophrastus carnifex, Abacophrastus hobbit, Abacophrastus megalops, Abacophrastus reflexus an...

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 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, 1} 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)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that ∑d i=1 fi(e)...

A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V rightarrow {0, 1, 2}$ suchthat every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = Sigma_{vin V} f(v)$The Roman domination number, $gamma_R(G)$, of $G$ is theminimum weight of an RDF on $G$.An RDF of minimum weight is called a $gamma_R$-function.A graph G is said to be $g...

Let k ≥ j ≥ 1 be two integers, and letG be a simple graph such that j(δ(G)+1) ≥ k, where δ(G) is the minimum degree of G. A (j, k)-dominating function of a graph G is a function f from the vertex set V (G) to the set {0, 1, 2, . . . , j} such that for any vertex v ∈ V (G), the condition ∑ u∈N[v] f(u) ≥ k is fulfilled, where N [v] is the closed neighborhood of v. A set {f1, f2, . . . , fd} of (j...

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...