Vertex Cover special cases
This is dominating set, not vertex cover.

The graph has the set of intervals as vertices and two vertices are adjacent if the corresponding intervals overlap. Such a graph is known as an interval graph, which is a special case of intersection graphs (vertices are sets of some kind, and two vertices are adjacent if their sets intersect).

You're looking for a minimum-sized set&nbsp;$S$ of intervals such that every other interval intersects at least one interval in&nbsp;$S$. In the interval graph, this means that you're looking for a minimum-sized set&nbsp;$S$ of vertices such that every other vertex is adjacent to&nbsp;$S$: this is a dominating set.

The problems of omputing minimum vertex covers and minimum dominating sets are both NP-hard on general graphs. However, on interval graphs, both problems can be solved in polynomial time, by


M.V. Marathe, R. Ravi and C. Pandu Rangan, Generalized vertex covering in interval graphs.  Discrete Applied Mathematics, 39(1):87–93, 1992.
K.S. Booth and J.H. Johnson, Dominating sets in chordal graphs. SIAM Journal on Computing, 11(1), 191–199, 1982.


In the former case, a $2$-vertex cover is an ordinary vertex cover; in the latter, the algorithm actually works for all for all directed-path graphs, and every interval graph is a directed-path graph.