  Let $(G,T)$ be an instance of the (vertex) multiway cut problem where $G$ is
a graph and $T$ is a set of terminals. For $t \in T$, a set of nonterminal
vertices separating $t$ from $T \setminus \{T\}$ is called an \emph{isolating
cut} of $t$. The largest among all the smallest isolating cuts is a natural
lower bound for a multiway cut of $(G,T)$. Denote this lower bound by $m$ and
let $k$ be an integer.
  In this paper we propose an $O(kn^{k+3})$ algorithm that computes a multiway
cut of $(G,T)$ of size at most $m+k$ or reports that there is no such multiway
cut. The core of the proposed algorithm is the following combinatorial result.
Let $G$ be a graph and let $X,Y$ be two disjoint subsets of vertices of $G$.
Let $m$ be the smallest size of a vertex $X-Y$ separator. Then, for the given
integer $k$, the number of \emph{important} $X-Y$ separators \cite{MarxTCS} of
size at most $m+k$ is at most $\sum_{i=0}^k{n \choose i}$.
