Definition:Connected (Graph Theory)/Graph

Definition

Let $G$ be a graph.

Then $G$ is connected if and only if every pair of vertices in $G$ is connected.

Disconnected

Let $G = \struct {V, E}$ be a graph.

Then $G$ is disconnected if and only if it is not connected.

That is, if and only if there exists (at least) two vertices $u, v \in V$ such that $u$ and $v$ are not connected.

Also see

• $k$-Connected

• Results about connected graphs can be found here.

Sources

• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): $\S 2.3$: Connected Graphs
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: walk
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: walk
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: connected graph