On the forcing connected domination number of a graph

摘要

By a connected simple graph G = (V, E), a subset D ? V is mentioned as connected dominating set of the graph G, if whose < D > is connected. The minimum cardinality of a connected dominating set is connected domination number. It is denoted by γ_c(G). For a minimum connected dominating set D of V a subset T ? D is called forcing subset for D if D is the unique minimum connected dominating set containing T. A forcing subset for D of minimum cardinality of a minimum forcing subset of D. The forcing connected domination number of D, denoted by f_(γc)(D), is the cardinality of a minimum forcing subset of D. The forcing connected domination number of G is denoted by f_(γc)(G), is f_(γc)(G) = min{f_(γc)(D)}, where the minimum is taken overall minimum connected dominating sets D in G. Some general properties satisfied by this concept are studied. The forcing connected domination number of certain classes of graphs are determined. Connected graphs of order p with forcing connected domination number 0 and 1 are characterized. Necessary and sufficient conditions for f_(γc)(G) be equal to γ(G) is given. We related the parameter domination number with connected domination number. We have the relation g (G) ≤ γ_c(G). We give the series of realization result for various possibilities of four parameters domination number, forcing domination number, connected domination number, forcing connected domination number.