  The square of a graph $G$, denoted $G^2$, is obtained from $G$ by putting an
edge between two distinct vertices whenever their distance is two. Then $G$ is
called a square root of $G^2$. Deciding whether a given graph has a square root
is known to be NP-complete, even if the root is required to be a chordal graph
or even a split graph.
  We present a polynomial time algorithm that decides whether a given graph has
a ptolemaic square root. If such a root exists, our algorithm computes one with
a minimum number of edges.
  In the second part of our paper, we give a characterization of the graphs
that admit a 3-sun-free split square root. This characterization yields a
polynomial time algorithm to decide whether a given graph has such a root, and
if so, to compute one.
