The　Min-Min　problem　of　finding　a　disjoint-path　pair　with　the　length　of　the　shorter　path　minimized　is　known　to　be　NP-hard　and　admits　no　K-approximation　for　any　K　＞　1　in　the　general　case　(Xu　et　al.　in　IEEE/ACM　Trans.　Netw.　14:147-158,　2006).　In　this　paper,　we　first　show　that　Bhatia　et　al.＇s　NP-hardness　proof　(Bhatia　et　al.　in　J.　Comb.　Optim.　12:83-96,　2006),　a　claim　of　correction　to　Xu　et　al.＇s　proof　(Xu　et　al.　in　IEEE/ACM　Trans.　Netw.　14:147-158,　2006),　for　the　edge-disjoint　Min-Min　problem　in　the　general　undirected　graphs　is　incorrect　by　giving　a　counter　example　that　is　an　unsatisfiable　3SAT　instance　but　classified　as　a　satisfiable　3SAT　instance　in　the　proof　of　Bhatia　et　al.　(J.　Comb.　Optim.　12:83-96,　2006).　We　then　gave　a　correct　proof　of　NP-hardness　of　this　problem　in　undirected　graphs.　Finally　we　give　a　polynomial-time　algorithm　for　the　vertex　disjoint　Min-Min　problem　in　planar　graphs　by　showing　that　the　vertex　disjoint　Min-Min　problem　is　polynomially　solvable　in　st-planar　graph　G=(V,E)　whose　corresponding　auxiliary　graph　G(V,Ea(a){e(st)})　can　be　embedded　into　a　plane,　and　a　planar　graph　can　be　decomposed　into　several　st-planar　graphs　whose　Min-Min　paths　collectively　contain　a　Min-Min　disjoint-path　pair　between　s　and　t　in　the　original　graph　G.　To　the　best　of　our　knowledge,　these　are　the　first　polynomial　algorithms　for　the　Min-Min　problems　in　planar　graphs.