Terminology: vertices in every cycle in a graph? Let $\mathcal{G} = (V, E)$ be a (directed or undirected)
graph. Call a set $S \subseteq V$ cycle-needed if every cycle in
$\mathcal{G}$ has at least one vertex in $S$. We say $S$ is minimally
cycle-needed if no cycle-needed set has a strictly smaller cardinality than $S$. In the example below the two  red nodes are cycle-needed but not minimally so, while
the yellow node is minimally cycle-needed.

What are    the 'official' names of cycle-neededness and minimal
cycle-neededness in mathematics? 
 A: A cycle-needed set is sometimes called a decycling set, and the size of a minimum cycle-needed set is called the decycling number of a graph (see The Decycling Numbers of Graphs by Bau and Beineke).
There's also terminology for the complementary notion: $S \subseteq V$ intersects every cycle of the graph $\mathcal G$ if and only if the subgraph induced by its complement $T = V \setminus S$ is a forest. As a result, looking for minimum decycling sets is equivalent to looking for maximum induced forests.
Somehow finding maximum induced trees in graphs has attracted more attention (there's a paper on this: Maximum Induced Trees in Graphs by Erdős, Saks, and Sós) but the problem is not equivalent even when $\mathcal G$ is connected. For example, in the butterfly graph

there is an induced forest of size $4$ (corresponding to the minimum decycling set of size $1$: the middle vertex) but only an induced tree of size $3$. But there are people talking about induced forests as well, especially in planar graphs (see this overview).
