Page 291 Exercise 3

According to the definition in the text, "a biconnected component of a graph G is a maximal binconnected subgraph H of G."  That is, "G contains no other subgraph that is both biconnected
and properly contains H."

Assume that we have an edge that is in two biconnected components.  This can occur only if a cycle is present.  However, cyclic graphs are not biconnected since removing an edge from a cyclic graph will not cause the graph to break into two distinct components.  Thus, we have a contradiction: If an edge lies in two distinct subgraphs, it must produce a cycle.  But cyclic graphs are not biconnected.