We consider an optimization problem that integrates network design and broadcast domination decisions. Given an undirected graph, a feasible broadcast domination is a set of nonnegative integer powers fi assigned to each node i, such that for any node j in the graph, there exists some node k having a positive fk-value whose shortest distance to node j is no more than fk. The cost of a broadcast...

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduced exponential domination, where vertices are considered to have some dominating power that decreases exponentially with the distance, and the dominated vertices have to accumulate a sufficient amount of this power emanating from the dominating vertice...

For a connected graph G = (V,E), a set D ⊆ V (G) is a dominating set of G if every vertex in V (G)−D has at least one neighbour in D. The distance dG(u, v) between two vertices u and v is the length of a shortest (u− v) path in G. An (u− v) path of length dG(u, v) is called an (u− v)-geodesic. A set X ⊆ V (G) is convex in G if vertices from all (a − b)-geodesics belong to X for any two vertices...

A broadcast on a graph G is a function f : V → {0, 1, 2, . . . }. The broadcast number of G is the minimum value of ∑ v∈V f(v) among all broadcasts f for which each vertex of G is within distance f(v) from some vertex v with f(v) ≥ 1. The broadcast number is bounded above by the radius and the domination number of G. We consider a class of trees that contains the caterpillars and characterize t...

The domination number γ(H) of a hypergraph H = (V (H), E(H)) is the minimum size of a subset D ⊂ V (H) of the vertices such that for every v ∈ V (H) \D there exist a vertex d ∈ D and an edge H ∈ E(H) with v, d ∈ H. We address the problem of finding the minimum number n(k, γ) of vertices that a k-uniform hypergraph H can have if γ(H) ≥ γ and H does not contain isolated vertices. We prove that n(...

We generalize to n steps the notion of exact 2-step domination introduced by Chartrand, et al in [2] and suggest a related minimization problem for which we nd a lower bound. A graph G is an exact n-step domination graph if there is some set of vertices in G such that each vertex in the graph is distance n from exactly one vertex in the set. We prove that such subsets have order at least blog2 ...

For an integer k ≥ 1, a (distance) k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V (G) \ S is at distance at most k from some vertex of S. The k-domination number, γk(G), of G is the minimum cardinality of a k-dominating set of G. In this talk, we establish lower bounds on the k-domination number of a graph in terms of its diameter, radius and gir...