  The cyclic shift graph of a monoid is the graph whose vertices are elements
of the monoid and whose edges link elements that differ by a cyclic shift. This
paper examines the cyclic shift graphs of `plactic-like' monoids, whose
elements can be viewed as combinatorial objects of some type: aside from the
plactic monoid itself (the monoid of Young tableaux), examples include the
hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search
trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary
search trees with multiplicities), and the Baxter monoid (pairs of twin binary
search trees). It was already known that for many of these monoids, connected
components of the cyclic shift graph consist of elements that have the same
evaluation (that is, contain the same number of each generating symbol). This
paper focusses on the maximum diameter of a connected component of the cyclic
shift graph of these monoids in the rank-$n$ case. For the hypoplactic monoid,
this is $n-1$; for the sylvester and taiga monoids, at least $n-1$ and at most
$n$; for the stalactic monoid, $3$ (except for ranks $1$ and $2$, when it is
respectively $0$ and $1$); for the plactic monoid, at least $n-1$ and at most
$2n-3$. The current state of knowledge, including new and previously-known
results, is summarized in a table.
