Let　G　be　a　graph　and　s>0　be　an　integer.　If,　for　any　function　b:V(G)→ℤ2s+1　satisfying　Σv∈V(G)b(v)≡0(mod2s+1),　G　always　has　an　orientation　D　such　that　the　net　outdegree　at　every　vertex　v　is　congruent　to　b(v)　mod　2s+1,　then　G　is　strongly　Z2s+1-connected.　For　a　graph　G,　denote　by　α(G)　the　cardinality　of　a　maximum　independent　set　of　G.　In　this　paper,　we　prove　that　for　any　integers　s,t>0　and　real　numbers　a,b　with　0<a<1,　there　exist　an　integer　N(a,b,s)　and　a　finite　family　Y(a,b,s,t)　of　non-strongly　ℤ2s+1-connected　graphs　such　that　for　any　connected　simple　graph　G　with　order　n≥N(a,b,s)　and　α(G)≤t,　if　G　satisfies　one　of　the　following　conditions:　for　any　edge　uv∈　E(G),　max{dG(u),　dGG(u),　dG2s+1-connected　if　and　only　if　G　is　not　contractible　to　a　member　in　the　finite　family　Y(a,b,s,t).　©　2015　Elsevier　B.V.　All　rights　reserved.