  A word is square-free 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 square-free words over a $3$-letter alphabet. We study square-free words
with additional properties involving single-letter deletions and extensions of
words.
  A square-free word is steady if it remains square-free 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 square-free words that remain
square-free 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 alphabet of size $12$.
