Linear-time recognition of bipartite graphs plus two edges

Journal article, 2003

Cai and Schieber (1997) proved that bipartite graphs plus one edge can be recognized in linear time. We extend their result to bipartite graphs plus two edges. Our algorithm works on a depth-first-search spanning tree. This gives, as a byproduct, also a simplified solution to the one-edge case. It is a notoriously open question whether recognizing bipartite graphs plus $k$ edges is a fixed-parameter tractable problem. The present result might support the affirmative conjecture.