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 ɛ 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 ∈ E(G), there is an independent set S in G such that |ΓG(S)| = |S| + 1, x ∈ S and |ΓG−xy(S)| = |S|. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\sqrt v \varepsilon )$\end{document} time that determines whether G is a perfect 2-matching covered graph or not.