On a conjecture of compatibility of multi-states characters

Perfect phylogeny consisting of determining the compatibility of a set of characters is known to be NP-complete [5, 30]. We propose in this article a conjecture on the necessary and sufficient conditions of compatibility: Given a set C of r-states full characters, there exists a function f(r) such that C is compatible iff every set of f(r) characters of C is compatible. According to [8, 10, 9, 27, 12, 25], f(2) = 2, f(3) = 3 and f(r) ≥ r− 1. [25] conjectured that f(r) = r for any r ≥ 2. In this paper, we present an example showing that f(4) ≥ 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).