Let　G　=　(V,　E)　be　a　connected　graph　and　P　be　graph-theoretic　property.　A　network　is　often　modeled　by　a　graph　G　=　(V,　E).　One　fundamental　consideration　in　the　design　of　networks　is　reliability.　The　connectivity　is　an　important　parameter　to　measure　the　fault　tolerance　and　reliability　of　network.　The　conditional　connectivity　lambda(G,　P)　or　kappa(G,　P)　is　the　minimum　cardinality　of　a　set　of　edges　or　vertices,　if　it　exists,　whose　deletion　disconnects　G　and　each　remaining　component　has　property　P.　Let　F　be　a　vertex　set　or　edge　set　of　G　and　P　be　the　property　of　with　at　least　k　components.　Then　we　have　the　k-component　connectivity　c　kappa　(k)(G)　and　the　k-component　edge　connectivity　c　lambda(k)(G).　In　this　paper,　we　determine　the　k-component　(edge)　connectivity　of　locally　twisted　cubes　LTQ(n)　for　small　k,　and　we　also　prove　other　properties　of　LTQ(n).　(C)　2018　Elsevier　Inc.　All　rights　reserved.