Let $F$ be a compact, orientable surface with negative Euler characteristic, 
and let $x_1, \cdots , x_n$ be $n$ fixed but arbitrarily chosen points on $\mathrm{int}F$, 
each of which has a (small) diskal neighborhood $D_i \subset F$. 
Denote by $\mathcal{S}_n(F)$ a subgroup of $\mathrm{Diff}(F)$ 
consisting of ``sliding" maps $f$ each of which satisfies \\ 
$(1)$ $f(\{x_1, \dots , x_n\}) = \{x_1, \dots , x_n\}$, 
$f(D_1\cup \cdots \cup D_n) = D_1 \cup \cdots \cup D_n$ and \\ 
$(2)$ $f$ is isotopic to the identity map on $F$. \\
Then by restricting such automorphisms to 
$\hat{F} = F - \mathrm{int}(D_1 \cup \cdots \cup D_n)$, 
we have automorphisms $\hat{f} : \hat{F} \to \hat{F}$, 
which form a subgroup $\mathcal{S}_n(\hat{F})$ 
of $\mathrm{Diff}(\hat{F})$. 
We give a Nielsen-Thurston classification of elements of 
$\mathcal{S}_n(\hat{F})$ using braids in $F \times I$ 
which characterize the elements of $\mathcal{S}_n(\hat{F})$.