A Negative Answer to a Question of Santos on the Dual Diameter of Surfaces with Boundary

Let f̄(2,n) be the maximum diameter of the dual graph of a triangulated surface with boundary on n vertices. At a 2012 Oberwolfach workshop, Santos asked whether f̄(2,n) ≤ n − 3. We show that the answer is no. A triangulation M₇ of the real projective plane minus two disjoint open discs has 7 vertices, 10 triangles and dual diameter 5. The same complex appears in work of Holmes on Stanley–Reisner rings with Serre's property (S₂), where it is not identified as a surface. Gluing copies of M₇ along boundary edges gives f̄(2,n) ≥ n − 3 + ⌊(n − 2)/5⌋ for all n ≥ 3, so the excess over n − 3 is unbounded. An orientable surface of genus 2 with 10 vertices and dual diameter 8 gives the bound n − 3 + ⌊(n − 2)/8⌋ for orientable surfaces. A short layering argument shows f̄(2,n) ≤ max(n − 3, 2n − 8), and a theorem of Holmes gives max(2n − 10, n − 2). Computer searches with DRAT-certified unsatisfiability proofs, confirmed by an exhaustive enumeration of the surfaces with at most 10 vertices, show that f̄(2,n) = n − 3 for n ≤ 6 and f̄(2,n) = n − 2 for 7 ≤ n ≤ 10 (for 6 ≤ n ≤ 9 these values also follow from results of Holmes), that M₇ is the only counterexample with at most 7 vertices, and that orientable surfaces with at most 9 vertices satisfy the bound. This is an unrefereed note.