Let k be a positive integer and G = (V,E) a connected graph of order n. A set D ⊆ V is called a distance k-dominating set of G if each x ∈ V (G)−D is within distance k from some vertex of D. The k-domination number of G, denoted by γk(G), is the minimum cardinality over all distance k-dominating sets. Determining γk(G) has a significant impact on an efficient design of routing protocols in netw...

In this paper, we study a variant of graph domination known as $(t, r)$ broadcast domination, first defined in Blessing, Insko, Johnson, and Mauretour 2015. variant, each provides $t-d$ reception to vertex distance $d < t$ from the broadcast. If \ge then no is provided. A considered dominated if it receives $r$ total all broadcasts. Our main results provide some upper lower bounds on density do...

Let G = (V, E) be a graph and let S ⊆ V . The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set S is a dominating set (DS) if every vertex in V − S is adjacent to a vertex in S. Further, if every vertex in V − S is also adjacent to a vertex in V − S, then S is a restrained dominating set (RDS). The domination number of G, denoted by γ(G), i...

The string-dominated universe locally resembles an open universe, and fits dynamical measures of power spectra, cluster abundances, redshift distortions, lensing constraints, luminosity and angular diameter distance relations and microwave background observations. We show examples of networks which might give rise to recent string-domination without requiring any fine-tuned parameters. We discu...

We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453–1462], where the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a t...

Suppose G = (V , E) is a simple graph and k is a fixed positive integer. A subset D ⊆ V is a distance k-dominating set of G if for every u ∈ V , there exists a vertex v ∈ D such that dG(u, v) ≤ k, where dG(u, v) is the distance between u and v in G. A set D ⊆ V is a distance k-paired-dominating set of G if D is a distance k-dominating set and the induced subgraph G[D] contains a perfect matchin...

Let nand k be positive integers and let G be a graph. A set D of vertices of G is defined to be an (n, k )-dominating set of G if every vertex of V( G) D is within distance n from at least k vertices of D. The minimum cardinality among all (n, k )-dominating sets of G is called the (n, k )-domination number of G and is denoted by 'Yn,k(G). A set I of vertices of G is defined to be an (n, k)inde...

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...

Distance-hereditary graphs are graphs in which every two vertices have the same distance in every connected induced subgraph containing them. This paper studies distance-hereditary graphs from an algorithmic viewpoint. In particular, we present linear-time algorithms for finding a minimum weighted connected dominating set and a minimum vertex-weighted Steiner tree in a distance-hereditary graph...