The class of cographs is one of the most well-known graph classes, which is also known to be equivalent to the class of P4-free graphs. We show that Mader's conjecture is true if we restrict ourselves to cographs, that is, for any tree T of order m, every k-connected cograph G
Nearby in the stack
with
δ(G)≥⌊23k⌋+m−1
contains a subtree
T′≅T
such that
G−V(T′)
is still
k
-connected, where
δ(G)
denotes the minimum degree of
G
. Moreover, we show that three variants of Mader's conjecture hold for cographs, that is, for any tree
T
of order
m
,
∙
every
k
-connected (respectively,
k
-edge-connected) cograph
G
with
δ(G)≥k+m−1
contains a subtree
T′≅T
such that
G−E(T′)
is
k
-connected (respectively,
k
-edge-connected),
∙
every
k
-edge-connected cograph
G
with
δ(G)≥k+m−[k=1]
contains a subtree
T′≅T
such that
G−V(T′)
is
k
-edge-connected, where we use Iverson's convention for
[k=1]
. We furthermore present tight lower bounds on the minimum degree of a cograph for the existence of disjoint connectivity keeping trees, a maximal connectedness keeping tree and a super edge-connectedness keeping tree.