Is a subgraph of G always connected

I am trying to figure out if given a connect graph with N nodes and A edges, its subgraphs are connected.

In order word: given a graph G, can I have a subgraph of G that is not connected? Or: can a subpgraph of G not be connected?

I think the answer is no because otherwise G' is not a (sub)graph.

• Graph theory has a different term for what you think is a subgraph: it's called a component. – reinierpost May 2 '19 at 8:59

A subgraph of a graph $$G$$ is any graph that has a subset of $$G$$'s vertices and a subset of its edges. In particular, for an graph $$G=(V,E)$$, the graph $$(V,\emptyset)$$ is a subgraph that's not connected.
• I'm saying a graph with no edges is a non-connected subgraph of $G$. A graph with exactly one edge is connected if it only has two vertices, and disconnected if it has more than two. – David Richerby May 1 '19 at 21:19
• @DavidRicherby the graph $(V,\emptyset)$ is a subgraph that's not connected except when $V$ is a set of a single vertex. – John L. May 2 '19 at 1:45