Generalisation of vertex neighborhood I am trying to find out what to call a generalized closed neighborhood of a vertex in an undirected graph, i.e. a subgraph containing all vertices up to a set distance from a selected "root" vertex, and the edges with both endpoints included in this set of vertices.
 A: Let $G=(V,E)$ be our graph, $v$ the root vertex and $k$ the distance. A common term for the vertex set of what you want is the ball of radius $k$ around $v$, denoted $B(v,k)$ or $B_k(v)$ or $N_k[v]$ depending on the author. So
$$B(v,k) = B_k(v) = N_k[v] = \{u\in V: d(u,v)\leq k\}$$
If you want to include the edges, then take the subgraph induced by this. So you might call it $G[B(v,k)]$, or just make your own notation if you're going to be using it a lot.
A: Some authors like to define and call this the distance-$k$ neighborhood of a vertex. If you do a search on e.g., Google Scholar you will find several papers using it. For example, one of my first hits is [1] with around 800 citations.
But indeed, this is only gives you the set of vertices at a distance at most $k$. You'd want to take the subgraph induced by this set; you might want to use your own definition to make the notation a bit lighter.

[1] Ahuja, R.K., Ergun, Ö., Orlin, J.B. and Punnen, A.P., 2002. A survey of very large-scale neighborhood search techniques. Discrete Applied Mathematics, 123(1-3), pp.75-102.
