  Perfect phylogeny consisting of determining the compatibility of a set of
characters is known to be NP-complete. We propose in this article a conjecture
on the necessary and sufficient conditions of compatibility: Given a set
$\mathcal{C}$ of $r$-states full characters, there exists a function $f(r)$
such that $\mathcal{C}$ is compatible iff every set of $f(r)$ characters of
$\mathcal{C}$ is compatible. Some previous work showed that $f(2)=2$, $f(3)=3$
and $f(r) \ge r-1$. Gusfield et al. 09 conjectured that $f(r) = r$ for any $r
\ge 2$. In this paper, we present an example showing that $f(4) \ge 5$ and then
a closure operation for chordal sandwich graphs. The later problem is a common
approach of perfect phylogeny. This operation can be the first step to simplify
the problem before solving some particular cases $f(4), f(5), ... $, and
determining the function $f(r)$.
