Algorithm for two disjoint long paths in 2-connected graphs

Denote by σ̄k = min {d(x1)+d(x2)+ · · ·+d(xk)−|N(x1)∩N(x2)∩· · ·∩N(xk)| | x1, x2, · · · , xk are k independent vertices in G}. Let n and m denote the number of vertices and edges of G. For any connected graph G, we give a polynomial algorithm in O(nm) time to either find two disjoint paths P1 and P2 such that |P1|+ |P2| ≥ min{σ̄4, n} or output G = ∪i=1Gi such that for any i, j ∈ {1, 2, ..., k} (k ≥ 3), V (Gi) ∩ V (Gj) = {v}, where v ∈ V (G).