Let gp t ( G ) , gp o ( G ) , and gp d ( G ) be the total, the outer, and the dual general position number of a graph G , respectively. This paper investigates how removing a vertex or removing an edge affects these graph invariants. It is proved that if x is not a cut vertex, then gp t ( G ) − 1 ≤ gp t ( G − x ) ≤ gp t ( G ) + deg G ( x ) . On the other hand, gp o ( G − x ) and gp d ( G − x ) can be respectively arbitrarily larger/smaller than gp o ( G ) and gp d ( G ) . On the positive side, it is proved that if x lies in some gp o -set, then gp o ( G ) − 1 ≤ gp o ( G − x ) , and that if x is not a cut vertex and lies in some gp d -set of G , then gp d ( G ) − 1 ≤ gp d ( G − x ) . For the edge removal, it is proved that (i) gp t ( G ) − | S ( G ) e | ≤ gp t ( G − e ) ≤ gp t ( G ) + 2 , where S ( G ) e is the set of simplicial vertices adjacent to both endvertices of e , (ii) gp o ( G ) / 2 ≤ gp o ( G − e ) ≤ 2 gp o ( G ) , and (iii) that gp d ( G ) − gp d ( G − e ) can be arbitrarily large. All bounds are demonstrated to be sharp.

The Maker–Breaker domination game is played on a graph G by two players, called Dominator and Staller, who alternately choose a vertex that has not been played so far. Dominator wins the game if his moves form a dominating set. Staller wins if she plays all vertices from a closed neighborhood of a vertex v∈V(G). Dominator’s fast winning strategies were studied earlier. In this work, we concentrate on the cases when Staller has a winning strategy in the game. We introduce the invariant γSMB′(G) (resp., γSMB(G)) which is the smallest integer k such that, under any strategy of Dominator, Staller can win the game by playing at most k vertices, if Staller (resp., Dominator) plays first on the graph G. We prove some basic properties of γSMB(G) and γSMB′(G) and study the parameters’ changes under some operators as taking the disjoint union of graphs or deleting a cut vertex. We show that the inequality δ(G)+1≤γSMB′(G)≤γSMB(G) always holds and that for every three integers r,s,t with 2≤r≤s≤t, there exists a graph G such that δ(G)+1=r, γSMB′(G)=s, and γSMB(G)=t. We prove exact formulas for γSMB′(G) where G is a path or it is a tadpole graph which is obtained from the disjoint union of a cycle and a path by adding one edge between them.