The　connectivity　of　a　connected　graph　G　is　the　minimum　cardinality　over　all　vertex-cuts,　which　can　be　determined　using　Menger＇s　theorem　(1927).　Then　G　is　super　connected　if　its　minimum　vertex-cut　is　always　composed　of　a　vertex＇s　neighborhood.　Two　generalized　extensions　to　this　classic　notion　of　connectivity　include　the　T-structure　connectivity　kappa(G;T)　and　the　T-substructure　connectivity　kappa(s)(G;　T),　for　which　T　is　the　given　structure　isomorphic　to　a　connected　subgraph　of　G.　Let　T*　denote　the　union　of　the　set　of　all　connected　subgraphs　of　T　and　the　set　of　the　trivial　graph.　In　this　article,　a　connected　graph　G　is　called　super　T-connected　if　the　minimum　degree　of　G　-　F　is　zero　for　each　minimum　T-cut　F　of　G;　analogously,　G　is　super　T*-connected　if　the　minimum　degree　of　G　-　F　is　zero　for　each　minimum　T*-cut　F　of　G.　Considering　the　n-dimensional　locally　twisted　cube　LTQn　with　n　＞=　3,　we　first　establish　both.(LTQ(n);　T)　and　kappa(s)(LTQ(n);　T)　and　then　determine　whether　LTQ(n)　is　super　T-connected　and　super　T*-　connected,　where　T　is　an　element　of　{K-1,K-1,　K-1,K-2,　K-1,K-3,　C-4}boolean　OR{P-k　vertical　bar　4　＜=　k　＜=　n}.　(C)　2021　Elsevier　B.V.　All　rights　reserved.