The Smallest 2-Terminal Graphs Beyond P5

For a graph G and a vertex v, let B_r(v) denote the closed ball of radius r centred at v. A connected graph is r-terminal if its radius-r balls are pairwise distinct, while deleting any one vertex destroys this property. Charon, Honkala, Hudry and Lobstein asked in 2007 whether a 2-terminal graph other than the five-vertex path P5 exists; the question was still listed as open in a 2024 survey. We give an explicit nine-vertex, eleven-edge, outerplanar subcubic example and verify it by a complete table of radius-two balls and deletion witnesses. An exhaustive computation over all connected unlabelled graphs of orders five through nine, combined with an easy argument for smaller orders, shows that no further example exists on at most eight vertices, and that exactly 99 nonisomorphic examples occur on nine vertices. Their edge counts are distributed as 10, 27, 29, 22, 9, 2 for 11, 12, 13, 14, 15, 16 edges, respectively. All 99 are planar, 63 are outerplanar, and the displayed graph is the unique subcubic example of order nine. The computation is accompanied by a library-free bitset verifier, a separate NetworkX implementation, graph6 certificates, source checksums, and complete positive-instance lists.