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

Polynomial time recognition of squares of ptolemaic graphs and 3-sun-free split graphs