A Linear Time Algorithm for Determining Almost Bipartite Graphs.

A graph G = (V, E) is called almost bipartite if G is not bipartite, but there exists a vertex v is an element of V such that G-{v} is bipartite. We consider the problem of testing if G is almost bipartite or not. This problem arises from the study on the k-arch layout problem. It is known that, given a graph G and an integer k >= 2, it is NP-complete to determine if G has a k-arch layout. On the other hand, G has a 1-arch layout if and only if G is almost bipartite [3]. It is straightforward to test if G is almost bipartite in O(n(n+ m)) time by using depth first search. In this paper, we present a simple linear time algorithm for solving this problem. The efficiency of the algorithm is achieved by sophisticated applications of depth first search tree and the study of the structure of such graphs.