Let \(k \ge 5\) be an odd integer and \(G=(V(G),E(G))\) be a k-edge-connected graph. For \(X\subseteq V(G)\) , e(X) denotes the number of edges between X and \(V(G)-X\) . We here prove that if \(\{s_i,t_i\}\subseteq X_i\subseteq V(G)\) , \(f_1\) is an edge between \(s_1\) and \(s_2\) , \(f_2\) is an edge between \(t_1\) and \(t_2\) , \(e(X_i)\le 2k-3\) \((i=1,2)\) , \(X_1\cap X_2=\emptyset\) , and \(e(Y)\ge k+2\) for each \(Y\subseteq V(G)\) with \(Y\cap \{s_1,t_1,s_2,t_2\}=\{s_1,t_2\}\) , then there exist paths \(P_1\) and \(P_2\) such that \(P_i\) joins \(s_i\) and \(t_i\) , \(V(P_i)\subseteq X_i\) ( \(i=1,2\) ) and \(G-\{f_1,f_2\}-E(P_1\cup P_2)\) is ( \(k-2\) )-edge-connected, and in fact we give a generalization of this result.
