Relating the annihilation number and the 2-domination number of a tree

A set S of vertices in a graph G is a 2-dominating set if every vertex of G not in S is adjacent to at least two vertices in S. The 2-domination number γ2(G) is the minimum cardinality of a 2-dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the nondecreasing degree sequence of G is at most the number of edges in G. The conjecture-generating computer program, Graffiti.pc, conjectured that γ2(G) ≤ a(G) + 1 holds for every connected graph G. It is known that this conjecture is true when the minimum degree is at least 3. The conjecture remains unresolved for minimum degree 1 or 2. In this paper, we prove that the conjecture is indeed true when G is a tree, and we characterize the trees that achieve equality in the bound. It is known that if T is a tree on n vertices with n1 vertices of degree 1, then γ2(T ) ≤ (n + n1)/2. As a consequence of our characterization, we also characterize trees T that achieve equality in this bound. Research supported in part by the University of Johannesburg and the South African National Research Foundation. Research supported in part by the University of Johannesburg.