The provisioning cost of a virtualized network (VN) depends on several factors, including the numbers of virtual routers (VRs) and virtual links (VLs), mapping of them on a substrate infrastructure, and routing of data traffic. An existing model, known as the virtual network embedding (VNE) model, determines the embedding of given VN graphs into the substrate infrastructure. When the resource allocation model of the VNE problem is adopted to a single-entity scenario, where a single entity fulfills the roles of both a service provider and an infrastructure provider, an issue of increased costs of VNs and access paths arise. This paper proposes a model for virtualized network graph design and embedding (VNDE) for the single-entity scenario. The VNDE model determines the number of VRs and a VN graph for each request in conjunction with embedding. The VNDE model also determines access paths that connect customer premises and VRs. We formulate the VNDE model as an integer linear programming (ILP) problem. We develop heuristic algorithms for the cases where the ILP problem cannot be solved in practical time. We evaluate the performance of the VNDE model on several networks, including an actual Japanese academic backbone network. Numerical results show that the proposed model designs suitable VN graphs and embeds them according to the volume of traffic demands and access path cost. Compared with the benchmark model, which is based on a classic VNE approach, the proposed model reduces the provisioning cost at most 28.7% in our examined scenarios.