  Let $\Lambda(T)$ denote the set of leaves in a tree $T$. One natural problem
is to look for a spanning tree $T$ of a given graph $G$ such that $\Lambda(T)$
is as large as possible. This problem is called maximum leaf number, and it is
a well-known NP-hard problem. Throughout recent decades, this problem has
received considerable attention, ranging from pure graph theoretic questions to
practical problems related to the construction of wireless networks.
  Recently, a similar but stronger notion was defined by Bradshaw,
Masa\v{r}\'ik, and Stacho [Flexible List Colorings in Graphs with Special
Degeneracy Conditions, ISAAC 2020]. They introduced a new invariant for a graph
$G$, called the robust connectivity and written $\kappa_\rho(G)$, defined as
the minimum value $\frac{|R \cap \Lambda (T)|}{|R|}$ taken over all nonempty
subsets $R\subseteq V(G)$, where $T = T(R)$ is a spanning tree on $G$ chosen to
maximize $|R \cap \Lambda(T)|$. Large robust connectivity was originally used
to show flexible choosability in non-regular graphs.
  In this paper, we investigate some interesting properties of robust
connectivity for graphs embedded in surfaces. We prove a tight asymptotic bound
of $\Omega(\gamma^{-\frac{1}{r}})$ for the robust connectivity of $r$-connected
graphs of Euler genus $\gamma$. Moreover, we give a surprising connection
between the robust connectivity of graphs with an edge-maximal embedding in a
surface and the surface connectivity of that surface, which describes to what
extent large induced subgraphs of embedded graphs can be cut out from the
surface without splitting the surface into multiple parts. For planar graphs,
this connection provides an equivalent formulation of a long-standing
conjecture of Albertson and Berman [A conjecture on planar graphs, 1979], which
states that every planar graph on $n$ vertices contains an induced forest of
size at least $n/2$.
