Answer
Showing that a simple graph is a tree if and only if it contains
no simple circuits and the addition of an edge connecting
two nonadjacent vertices produces a new graph that has
exactly one simple circuit.
Work Step by Step
Suppose T is a tree.
- Then clearly T has no simple circuits.
-If we add an edge e connecting two nonadjacent vertices u and
v, then obviously a simple circuit is formed,
-because when e
is added to T the resulting graph has too many edges to be a
tree.
- The only simple circuit formed is made up of the edge e
together with the unique path in T from v to u. Suppose T satisfies
the given conditions.
-- All that is needed is to show that T is connected, because there are no simple circuits in the graph.
--Assume that T is not connected. Then let u and v be in separate
connected components. Adding e = {u, v} does not satisfy
the conditions.
