Extensions and reductions of squarefree words

A word is squarefree if it does not contain a nonempty word of the form XX as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily long squarefree words over a 3-letter alphabet. We study squarefree words with additional properties involving single-letter deletions and extensions of words. A squarefree word is steady if it remains squarefree after deletion of any single letter. We prove that there exist infinitely many steady words over a 4-letter alphabet. We also demonstrate that one may construct steady words of any length by picking letters from arbitrary alphabets of size 7 assigned to the positions of the constructed word. We conjecture that both bounds can be lowered to 4, which is best possible. In the opposite direction, we consider squarefree words that remain squarefree after insertion of a single (suitably chosen) letter at every possible position in the word. We call them bifurcate. We prove a somewhat surprising fact, that over a fixed alphabet with at least three letters, every steady word is bifurcate. We also consider families of bifurcate words possessing a natural tree structure. In particular, we prove that there exists an infinite tree of doubly-infinite bifurcate words over an alphabet of size 12.