Given a model of a system and an objective, the model-checking problem asks whether the model satisfies the objective. In this case, the model is a standard graph, and the objective is safety: given a set of target vertices $T\subseteq V$, determine whether or not there is a path that does not visit any vertex in $T$ (i.e. you want to avoid all vertices in $T$). Furthermore, in the disjunctive queries problem, you are given multiple safety objectives (i.e. multiple target sets $T_i$) and you are to determine whether or not there is a path where at least one of the safety objectives is satisfied (i.e. whether or not an infinite path avoids all vertices in at least one of the target sets). Disjunctive queries coincide with disjunctive objectives on graphs.