SOLVABLE Graphs (also known as Reachable Graphs) are types of graphs that any arrangement of a specified number of agents located on the graph’s vertices can be reached from any initial arrangement through agents’ moves along the graph’s edges, while avoiding deadlocks (interceptions). In this paper, the properties of SOLVABLE Graphs are investigated, and a new concept in multi agent motion planning, called Minimal SOLVABLE Graphsis introduced. Minimal SOLVABLE Graphs are the smallest graphs among SOLVABLE Graphs in terms of the number of vertices. Also, for the first time, the problem of deciding whether a graph is SOLVABLE form agents is answered, and a new algorithm is presented for making an existing graph SOLVABLE and lean for a given number of agents. Finally, through an industrial example, it is demonstrated that how the findings of this paper can be used in designing and reshaping transportation networks (e.g. railways, traffic roads, AGV routs, robotic workspaces, etc.) for multiple moving agents such as trains, vehicles, and robots.