Based on the tile method for DNA self-assembly, which involves branched junction molecules whose flexible k-arms are double strands of DNA, we design a collection of tiles that will construct a nanostructure shaped like a target graph G. We find the minimum number of tile and bond-edge types required to construct friendship graphs and dumbbell-friendship graphs, which join two friendship graphs that have an extra edge that connects the common vertices of each friendship graph, in three different scenarios representing different levels of laboratory constraints.