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The binary perfect phylogeny model is too restrictive to model biologicalevents such as back mutations. In this paper we consider a naturalgeneralization of the model that allows a special type of back mutation. Weinvestigate the problem of reconstructing a near perfect phylogeny over abinary set of characters where characters are persistent: characters can begained and lost at most once. Based on this notion, we define the problem ofthe Persistent Perfect Phylogeny (referred as P-PP). We restate the P-PPproblem as a special case of the Incomplete Directed Perfect Phylogeny, calledIncomplete Perfect Phylogeny with Persistent Completion, (refereed as IP-PP),where the instance is an incomplete binary matrix M having some missingentries, denoted by symbol ?, that must be determined (or completed) as 0 or 1so that M admits a binary perfect phylogeny. We show that the IP-PP problem canbe reduced to a problem over an edge colored graph since the completion of eachcolumn of the input matrix can be represented by a graph operation. Based onthis graph formulation, we develop an exact algorithm for solving the P-PPproblem that is exponential in the number of characters and polynomial in thenumber of species.

The Binary Perfect Phylogeny with Persistent characters