On a Conjecture about Compatibility of Multi-states Characters

Perfect phylogeny consisting of determining the compatibility of a set of characters is known to be NP-complete [4,28]. 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 [7,9,8,25,11,23], f(2) = 2, f(3) = 3 and f(r) ≥ r. [23] conjectured that f(r) = r for any r ≥ 2. In this paper, we present an example showing that f(4) ≥ 5. Therefore it could be the case that for r ≥ 4 characters the problem behavior drastically changes. In a second part, we propose a closure operation for chordal sandwich graphs. The later problem is a common approach of perfect phylogeny.