Definition of vertex in graph theory

In all definitions of graph I know of (undirected graph, simple graph, directed graph, multigraph, hypergraph) the vertices are dedicated part of the data, ie. in all these cases you start with a set $V$ of vertices, which is then turned into a graph by attaching edges from a set $E$ to these vertices.

Sometimes you can recover the vertices from the set of edges, for example when you have functions $s,t:E rightarrow V$ giving to each edge a source- and a target vertex.

General points of an edge are usually not considered to be a vertex, so if there are finitely many edges you will have finitely many vertices. Yet it is possible to have $V$ being an infinite set (e.g. $Bbb Z=: V$ considered as a graph with an edge between each $n$ and $n+1$) or even $V$ finite and $E$ infinite (e.g. Hawaiian earrings). The standard assumption of combinatorics/graph theory is that $V$ and $E$ are assumed to be finite, though.