Domination and leaf density in graphs

The domination number γ(G) of a graph G is the minimum cardinality of a subset D of V (G) with the property that each vertex of V (G) − D is adjacent to at least one vertex of D. For a graph G with n vertices we define ǫ(G) to be the number of leaves in G minus the number of stems in G, and we define the leaf density ζ(G) to equal ǫ(G)/n. We prove that for any graph G with no isolated vertex, γ(G) ≤ n(1−ζ(G))/2 and we characterize the extremal graphs for this bound. Similar results are obtained for the total domination number. The 2-partition domination number γ(G, π2) of a graph G and a 2-partition π2 = {V1, V2} of V (G) is defined by the sum γ(G) + γG(V1)+γG(V2). We prove that for any graph G with no isolated vertex and any 2-partition π2 of V (G), γ(G, π2) ≤ 3n(1−ζ(G))/2 and we characterize the extremal graphs. For graphs with leaf density ζ > 1/6, this new bound is an improvement of the bound given by B. Hartnell and P. D. Vestergaard [J. Combin. Math. Combin. Comput. 2003, Aug., 46].