Some properties of 3-domination-critical graphs Original Research Article

A graph G is 3-γ-critical if its domination number γ is 3 and the addition of any edge decreases γ by 1. Wojcicka conjectured that every 3-γ-critical graph with minimum degree δ⩾2 has a hamiltonian cycle. In this paper, we prove that if G is a 3-γ-critical connected graph of order n with minimum degree δ⩾2, then (1) G is 1-tough; (2) the circumference of G is at least n−1.