Connected domination of regular graphs

Connected domination of regular graphs

A dominating set D of a graph G is a subset of V (G) such that for every vertex v ∈ V (G), either v ∈ D or there exists a vertex u ∈ D that is adjacent to v in G. Dominating sets of small cardinality are of interest. A connected dominating set C of a graph G is a dominating set of G such that the subgraph induced by the vertices of C in G is connected. A weakly-connected dominating set W of a g...

Generalized connected domination in graphs

As a generalization of connected domination in a graph G we consider domination by sets having at most k components. The order γ c (G) of such a smallest set we relate to γc(G), the order of a smallest connected dominating set. For a tree T we give bounds on γ c (T ) in terms of minimum valency and diameter. For trees the inequality γ c (T ) ≤ n− k − 1 is known to hold, we determine the class o...

Minus domination in regular graphs

A three-valued function f defined on the vertices of a graph G = (V,E), f : V , ( 1 , 0 , 1), is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v E V, f(N[v])>~ 1, where N[v] consists of v and every vertex adjacent to v. The weight of a minus dominating function is f ( V ) = ~ f (v) , over all vertices v E V. The mi...

Decycling connected regular graphs

On Distance Connected Domination Numbers of Graphs

Let k be a positive integer and G = (V,E) be a connected graph of order n. A set D ⊆ V is called a k-dominating set of G if each x ∈ V (G) − D is within distance k from some vertex of D. A connected k-dominating set is a k-dominating set that induces a connected subgraph of G. The connected k-domination number of G, denoted by γ k(G), is the minimum cardinality of a connected k-dominating set. ...