Name of particular graph theory vertex cover Given a graph $G$ with vertex set $V$, what is the name of a vertex cover $V'$ such that all vertices in $V$ are either in $V'$ or adjacent to a vertex in $V'$? (Is this type of cover even already defined? It's been years since I've thought hard about graph theory.) For a particular graph, I'm looking for a minimally sized cover of this type.
 A: A vertex cover is already a technical term. A set $V' \subseteq V$ is a vertex cover if every edge of the graph has at least one endpoint of $V'$. 
If you mean "what is the name for a vertex cover with the additional property that every vertex is either an element of $V'$ or has a neighbor in $V'$?" then there is no name for this concept, but it's very close to just a vertex cover. If the graph has no isolated vertices, then every vertex cover already has this property: if $V'$ is a vertex cover and $v$ is a vertex outside $V'$, then let $vw$ be any edge out of $v$; it must be covered by $V'$, so we must have $w \in V'$, and therefore $v$ has a neighbor in $V'$. 
If the graph has isolated vertices, then such a vertex cover is just any vertex cover that contains all of those isolated vertices.

If you did not mean the term "vertex cover" in its technical sense, and you're just looking for a set $V' \subseteq V$ such that every vertex is either an element of $V'$ or has a neighbor in $V'$, then what you're looking for is a dominating set.

In either case, the problem of finding the smallest such set is a well-known NP-complete problem, so there are no efficient algorithms for doing it exactly, although approximation algorithms are known.
