Bipartite double cover and perfect 2-matching covered graph with its algorithm

Let B(G) denote the bipartite double cover of a non-bipartite graph G with v >= 2 vertices and epsilon edges. We prove that G is a perfect 2-matching covered graph if and only if B(G) is a 1-extendable graph. Furthermore, we prove that B(G) is a minimally 1-extendable graph if and only if G is a minimally perfect 2-matching covered graph and for each e = xy is an element of E(G), there is an independent set S in G such that vertical bar Gamma(G)(S)vertical bar = vertical bar S vertical bar + 1, x is an element of S and vertical bar Gamma(G-xy)(S)vertical bar = vertical bar S vertical bar. Then, we construct a digraph D from B(G) or G and show that D is a strongly connected digraph if and only if G is a perfect 2-matching covered graph. So we design an algorithm in O(root v epsilon) time that determines whether G is a perfect 2-matching covered graph or not.