the broadcast domination number of the cartesian product of two cycles is completely determined.

The domination number of a graph G = (V,E) is the minimum cardinality of any subset S ⊂ V such that every vertex in V is in S or adjacent to an element of S. Finding the domination numbers of m by n grids was an open problem for nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomassé. Many variants of domination number on graphs, such as double domination number a...

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 broadcast domination problem of a graph G = (V,E) is to find a subset B ⊆ V such that the vertices in V \B can be dominated by some vertex in B. The difference between broadcast domination and classical domination is that each vertex v in B is assigned a power value f(v) where f(v) ≥ 1 and the vertices in distance at most f(v) to v can hear (be dominated by) v. The goal is to find the minim...

Broadcast domination assigns an integer value f(u) 0 to each vertex u of a given graph, such that every vertex u with f(u) = 0 is within distance f(v) from a vertex v with f(v) > 0. We can regard the vertices v with f(v) > 0 as broadcast stations, each having a transmission power that might be di erent from the powers of other stations. The optimal broadcast domination problem seeks to minimize...

In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical domination where selected vertices may have different domination powers. The minimum cost of a dominating broadcast in a graph G is denoted γb(G). The dual of this problem is called multipacking : a multipacking is a set M ⊆ V (G) such that for any vertex v and any positive integer r, the ball of radius r aro...

Broadcast domination was introduced by Erwin in 2002, and it is a variant of the standard dominating set problem, such that vertices can be assigned various domination powers. Broadcast domination assigns a power f(v) 0 to each vertex v of a given graph, such that every vertex of the graph is within distance f(v) from some vertex v having f(v) 1. The optimal broadcast domination problem seeks t...

Broadcast domination was introduced by Erwin in 2002, and it is a variant of the standard dominating set problem, such that different vertices can be assigned different domination powers. Broadcast domination assigns an integer power f(v) ≥ 0 to each vertex v of a given graph, such that every vertex of the graph is within distance f(v) from some vertex v having f(v) ≥ 1. The optimal broadcast d...

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...

a dominating set $d subseteq v$ of a graph $g = (v,e)$ is said to be a connected cototal dominating set if $langle d rangle$ is connected and $langle v-d rangle neq phi$, contains no isolated vertices. a connected cototal dominating set is said to be minimal if no proper subset of $d$ is connected cototal dominating set. the connected cototal domination number $gamma_{ccl}(g)$ of $g$ is the min...