Definition:Vertex Deletion

Definition
Let $G = \left({V, E}\right)$ be an (undirected) graph.

Let $W \subseteq V$ be a set of vertices of $G$.

Then the graph obtained by deleting $W$ from $G$, denoted by $G - W$, is the subgraph induced by $V \setminus W$.

Alternatively:
 * $G - W = \left({V \setminus W, \left\{{e \in E : e \cap W = \varnothing}\right\}}\right)$

Informally, $G - W$ is the graph obtained from $G$ by removing all vertices in $W$ and all edges incident to those vertices.

If $W$ is a singleton such that $W = \left\{{v}\right\}$, then $G - W$ may be expressed $G - v$.

Also see

 * Definition:Edge Deletion


 * Definition:Vertex Cut, where the vertex deletion separates the graph into disconnected components.
 * Definition:Cut-Vertex, which is a singleton Vertex Cut.