Relating the annihilation number and the 2-domination number for unicyclic graphs

The 2-domination number gamma 2(G) of a graph G is the minimum cardinality of a set S subset of V(G) such that every vertex from V(G) \ S is adjacent to at least two vertices in S. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of its edges. It was conjectured that gamma 2(G) <= a(G) + 1 holds for every non-trivial connected graph G. The conjecture was earlier confirmed for graphs of minimum degree 3, trees, block graphs and some bipartite cacti. However, a class of cacti were found as counterexample graphs recently by Yue et al. [9] to the above conjecture. In this paper, we consider the above conjecture from the positive side and prove that this conjecture holds for all unicyclic graphs.