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　κ(G;T)　and　the　T-substructure　connectivity　κs(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　κ(LTQn;T)　and　κs(LTQn;T)　and　then　determine　whether　LTQn　is　super　T-connected　and　super　T-connected,　where　T∈{K1,1,K1,2,K1,3,C4}∪{Pk|4≤k≤n}.　©　2021　Elsevier　B.V.