Vertex-minimal graphs with dihedral symmetry I

Let D2n denote the dihedral group of order 2n, where n≥3. In this article, we build upon the findings of Haggard and McCarthy who, for certain values of n, produced a vertex-minimal graph with dihedral symmetry. Specifically, Haggard considered the situation when n2 or n is a prime power, and McCarthy investigated the case when n is not divisible by 2, 3 or 5. In this article, we assume n is not divisible by 4 and construct a vertex-minimal graph whose automorphism group is isomorphic to D2n.