CONNECTED ETERNAL DOMINATION IN GRAPHS
Eternal dominating set (eternal secure set) of a graph G is a dominating set D of G such that, for any possible sequence of vertices there exists a sequence of dominating sets satisfying the conditions where and for is possible). The cardinality of a smallest eternal secure set of G is the eternal domination number or the eternal security number. A general version of eternal dominating set is called an eternal m-secure set. In this paper, we define a connected eternal m-secure set, which is an eternal m-secure set, whose induced subgraph is connected. The connected eternal m-security number of G is the minimum of the cardinalities of all connected eternal m-secure sets of the graph. Connected eternal m-security number of some classes of graphs is also determined.
eternal 1-secure set, eternal m-secure set, connected eternal m-secure set, connected eternal m-security number.