The shortest path distances obey Ptolemy's inequality: for every four vertices,,, and, the inequality holds. For instance, the gem graph in the illustration is not Ptolemaic, because in this graph, greater than.
For every two overlapping maximal cliques, the intersection of the two cliques is a separator that splits the differences of the two cliques. In the illustration of the gem graph, this is not true: cliques and are not separated by their intersection,, because there is an edge that connects the cliques but avoids the intersection.
The graph is both chordal and distance-hereditary. The gem shown is chordal but not distance-hereditary: in the subgraph induced by, the distance from to is 3, greater than the distance between the same vertices in the whole graph. Because both chordal and distance-hereditary graphs are perfect graphs, so are the Ptolemaic graphs.
The graph is chordal and does not contain an induced gem, a graph formed by adding two non-crossing diagonals to a pentagon.
The graph is distance-hereditary and does not contain an induced 4-cycle.
The graph can be constructed from a single vertex by a sequence of operations that add a new degree-one vertex, or duplicate an existing vertex, with the exception that a twin operation in which the new duplicate vertex is not adjacent to its twin can only be applied when the neighbors of the twins form a clique. These three operations without the exception form all distance-hereditary graph. To form all Ptolemaic graphs, it is not enough to use pendant vertices and true twins; the exceptional case of false twins is sometimes also required.
The convex subsets of vertices form a convex geometry. That is, every convex set can be reached from the whole vertex set by repeatedly removing an extreme vertex, one that does not belong to any shortest path between the remaining vertices. In the gem, the convex set cannot be reached in this way, because neither nor is extreme.
Based on the characterization by oriented trees, Ptolemaic graphs can be recognized in linear time.
Enumeration
The generating function for Ptolemaic graphs can be described symbolically, allowing the rapid calculation of the numbers of these graphs. Based on this method, the number of Ptolemaic graphs with labeled vertices, for, has been found to be
