Characteristics of Finite Tree/Condition 4

Theorem

Then:

$T$ is a finite tree if and only if $T$ has no circuits, but adding one edge creates a cycle.

Let $T = \struct {V, E}$ be a finite tree.

Then by definition $T$ is connected and has no circuits.

Let $u, v \in V$ be any two vertices of $T$.

Let $P = \tuple {u, u_1, u_2, \ldots, u_{n - 1}, v}$ be a path from $u$ to $v$.

Let a new edges $\set {u, v}$ be added.

Then $\tuple {u, u_1, u_2, \ldots, u_{n - 1}, v, u}$ is now a cycle, which is by definition also a circuit, in $T$.

Note that this applies even when $P = \tuple {u, v}$: $\tuple {u, v, u}$ is still a cycle in $T$, but now $T$ is a multigraph.

$\Box$

Suppose $T$ has no circuits, but adding one edge creates a cycle, which is by definition also a circuit.

If $T$ were disconnected, then it would be possible to add an edge $e$ to connect two components of $T$.

By definition, $e$ would be a bridge.

From Condition for Edge to be Bridge, it follows that $e$ does not lie on a circuit.

So, if the only way to add an edge to $T$ forms a cycle, it follows that $T$ must be connected.

So $T$ is connected and has no circuits.

Hence by definition $T$ is a tree.

$\blacksquare$