Let G = (V, E) be a simple graph. A function ƒ : V→ {- 1,1} is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of G, denoted by [formula], equals to the maximum weight of an inverse signed total dominating function of G. In this paper, we establish upper bounds on the inverse signed total domination number of graphs in terms of their order, size and maximum and minimum degrees.
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 ƒ: E(G) →{−1, 1} is said to be a signed star k-dominating function on [formula] 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 [formula] for each e ∈ E(G) is called a signed star (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (k, k)-dominating family on G is the signed star (k, k)-domatic number of G, denoted by [formula]. In this paper we study properties of the signed star (k, k)-domatic number [formula]. In particular, we present bounds on [formula], and we determine the signed (k, k)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213–218] for the signed star domatic number.
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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 Σe∈E(v) ≥ 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 y{k}ss(G) = min{ Σe∈Ef(v) | f is a SS{k}DF on G}. A set {f1, f2,..., fd} of distinct signed star {k}-dominating functions on G with the property that …[wzór] 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 y{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.
Let k ≥ 0 be an integer. A set S of vertices of a graph G = (V (G), E(G)) is called a global offensive k-alliance if /N(v) ∩ S/ ≥ /N(v) - S/ + k for every v ∈ V (G) - S, where 0 ≤ k Δ and Δ is the maximum degree of G. The global offensive k-alliance number [formula] is the minimum cardinality of a global offensive k-alliance in G. We show that for every bipartite graph G and every integer k ≥ 2, [formula], where Lk(G) is the set of vertices of degree at most k - 1. Moreover, extremal trees attaining this upper bound are characterized.
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