Relating the annihilation number and the 2-domination number of block graphs

Abstract The 2-domination number γ 2 ( G ) of a graph G is the order of a smallest set D ⊆ V ( G ) such that each vertex of V ( G ) ∖ D is adjacent to at least two vertices in D . The annihilation number a ( G ) of G is the largest integer k such that there exist k different vertices in G with degree sum of at most | E ( G ) | . It is conjectured that γ 2 ( G ) ≤ a ( G ) + 1 holds for every nontrivial connected graph G . The conjecture was proved for graphs with minimum degree at least 3, and remains unresolved for graphs with minimum degree 1 or 2. In this paper we establish the conjecture for block graphs.