In this paper we study domination number of Johnson graphs J (n, r). We establish general bounds for the domination number of Johnson graph J(n, r) and prove that it is non-decreasing with n. We also compute exact values for γ(J(n, r)) in specific cases, showing that γ(J(n,2)) = ⌊n/2⌋ for n ≥ 4, and we determine γ(J(n , 3)) for even n ≥ 6, which depends quadratically on n. For odd n ≥ 7, we provide an upper bound for γ(J(n , 3)), which we conjecture to be tight. Our approach relies on translating the problem of computing γ(J(n , 3)) into the problem of finding a minimum edge covering by triangles among all graphs of order n with independence number at most 2.

The Maker–Breaker domination game (MBD game) is a two-player game played on a graph G by Dominator and Staller. They alternately select unplayed vertices of G . The goal of Dominator is to form a dominating set with the set of vertices selected by him while that of Staller is to prevent this from happening. In this paper MBD game critical graphs are studied. Their existence is established and critical graphs are characterized for most of the cases in which the first player can win the game in one or two moves.