On Avoidable Two Element Sets of Partial Words∗

This paper considers the problem of finding avoiding words given sets of partial words of the form {a 1a m2 . . . ka, b 1b n2 . . . lb} where a and b are letters in the alphabet and is a hole, or showing the set is unavoidable. Such a set is avoidable if and only if there exists a two-sided infinite full word with no factor compatible with a member of the set, and this word is called the avoiding word. For the case of k = 1, l = 1, we identify a strict minimum period for an avoiding word. For the case k = 1, l = 2, we refine a conjecture which was identified by Blanchet-Sadri et al. that, if proven, suffices to classify all two element sets [4]. We also discover a collection of unavoidable sets for the case k = 1, l = 3, and correct a previous result, which claimed no such sets exist. Finally, given our refined conjecture, we attempt to classify the remaining sets by identifying common avoiding words and exhibiting exactly which sets these words can avoid. In the process, we introduce a new technique to reduce a set if the period of the avoiding word is known. Using this technique, we are able to classify a significant number of sets previously unclassified.