2013

Fasciculi Mathematici

Rocznik: 2013 | Numer: nr 51

artykuł naukowy

angielski

EN 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 Σ_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{Σ_e∈Ef(e) | f is a SS{k}DF on G}. A set {f₁, f₂,..., f_d} of distinct signed star {k}-dominating functions on G with the property that ∑_i=1^d f_i (e)≤ k for each e ∈ E(G), is called a signed star {k}-dominating family (of functions) on G. The maximum number of functions in a signed star {k}-dominating family on G is the signed star {k}-domatic number of G, denoted by d_{k}SS(G). In this paper we study the properties of the signed star {k}- domination number γ_{k}SS(G) and signed star {k}-domatic number d_{k}SS(G). In particular, we determine the signed star {k}-domination number of some classes of graphs. Some of our results extend these one given by Xu [7] for the signed star domination number and Atapour et al. [1] for the signed star domatic number.

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