Question

Show that a graph is not orientable if it has a cut edge.

Short answer

It is proved that the graph having a cut edge can’t be orientable.

Step-by-step solution

Concept/Significance of orientable graph

If there is an orientation (a set of arc directions) that makes D strongly linked and there is a directed path between every pair of vertices, the graph is said to be orientable.

Determination of whether graph is orientable or not

Assume that the undirected graph G has a cut edge a, b.

When the edge is eliminated, a and b become independent components of G; in other words, all paths from a to b must pass through the edge in the a to b direction, and all paths from b to a must pass through the route in the b to a direction.

Assume that we know the graph's orientation.

If a, bare oriented as (a, b), then there can't be a path from b to ain the resulting directed graph.

Hence, the resulting directed graph isn't strongly linked, as observed.

If a, b, on the other hand, is orientated as (b, a), then there can't be a path from a tob in the resultant directed graph.

As a result, G is not orientable by definition.

Intriguingly, the inverse of this conclusion is also true.

Thus, it is proved that the graph having a cut edge can’t be orientable.