Let S⊆V(G) and κG​(S) denote the maximum number k of
edge-disjoint trees T1​,T2​,⋯,Tk​ in G such that
V(Ti​)â‹‚V(Tj​)=S for any i,j∈{1,2,⋯,k} and iî€ =j. For an integer r with 2≤r≤n, the {\em generalized
r-connectivity} of a graph G is defined as κr​(G)=min{κG​(S)∣S⊆V(G) and ∣S∣=r}. The r-component
connectivity cκr​(G) of a non-complete graph G is the minimum number
of vertices whose deletion results in a graph with at least r components.
These two parameters are both generalizations of traditional connectivity.
Except hypercubes and complete bipartite graphs, almost all known
κr​(G) are about r=3. In this paper, we focus on κ4​(Dn​)
of dual cube Dn​. We first show that κ4​(Dn​)=n−1 for n≥4.
As a corollary, we obtain κ3​(Dn​)=n−1 for n≥4. Furthermore,
we show that cκr+1​(Dn​)=rn−2r(r+1)​+1 for n≥2 and
1≤r≤n−1