etant donné le rapport réciproque entre la société et la littérature, et vu la dominance extraordinaire du roman, à l'état actuel, sur les autres expressions littéraires, on ne peut s'empêcher de s'interroger sur la cause et l'origine de la primauté du genre romanesque. certes, le roman ne date pas du xixe siècle; il est l'un des héritages des siècles précédents. néanmoins, son déploiement est ...

A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed dominating function if for any vertex v the sum of function values over its closed neighborhood is at least 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. By simply changing “{+1,−1}” in the above definition to “{+1, 0,−1}”, we can define the minus dominating f...

Let G = (V,E) be a simple graph. For any real function g : V −→ R and a subset S ⊆ V , we write g(S) = ∑ v∈S g(v). A function f : V −→ [0, 1] is said to be a fractional dominating function (FDF ) of G if f(N [v]) ≥ 1 holds for every vertex v ∈ V (G). The fractional domination number γf (G) of G is defined as γf (G) = min{f(V )|f is an FDF of G }. The fractional total dominating function f is de...

A three-valued function f defined on the vertices of a graph G = (V,E), f : V , ( 1 , 0 , 1), is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v E V, f(N[v])>~ 1, where N[v] consists of v and every vertex adjacent to v. The weight of a minus dominating function is f ( V ) = ~ f (v) , over all vertices v E V. The mi...

Let $G=(V,E)$ be a finite and simple graph of order $n$ maximumdegree $\Delta$. A signed strong total Roman dominating function ona $G$ is $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ open neighborhood (ii) every forwhich $f(v)=-1$ adjacent to at least one vertex...

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N−[v] f(x) ≥ k for each v ∈ V (D), where N−[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set {f1, f2, . . . , fd} of distinct signed k-dominating functions of D with the property tha...

Let G be a 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 distinct signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ k for each x ∈ V (G), is call...

For a positive integer k, a k-rainbow dominating function of a digraph D is a function f from the vertex set V (D) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (D) with f(v) = ∅ the condition u∈N−(v) f(u) = {1, 2, . . . , k} is fulfilled, where N−(v) is the set of in-neighbors of v. A set {f1, f2, . . . , fd} of k-rainbow dominating functions on D with t...

Let. n ?: 1 be an integer and lei G = (V, E) be a graph. In this paper we study a non discrete generalization of l'n(G), the maximum cardinality of a minimal n-dominating sei in G. A real-valued function f : V -t [0,1] is n-dominating if for each v E V, the sum of the values assigned to the vertices in the closed n-neighbourhood of v, Nn[v], is at least one, i.e., f(Nn [ll]) ?: 1. The weight of...