A linear-time algorithm for finding an ambitus

We devise a linear-time algorithm for finding an ambitus ín an undirected graph. An ambitus is a cycle in a graph containing two distinguished vertices such that certain different groups of bridges (called B^itp-, B^itQ-, and B^itPQ-bridges) satisfy the property that a bridge in one group does not interlace with any bridge in the other groups. Thus, an ambitus allows the graph to be cut into pieces, where, in each piece, certain graph properties may be investigated independently and recursively, and then the pieces can be pasted together to yield information about these graph properties in the original graph. In order to achieve a good time-complexity for such an algorithm employing the divide-and-conquer paradigm, it is necessary to find an ambitus quickly. We also show that, using ambitus, linear-time algorithms can be devised for abiding-path-finding and nonseparating-induced-cycle-finding problems.