# complexity theory – Trapezoidal Decomposition on A Graph

I know that when we plan the motion of a robot we may apply the trapezoidal decomposition of free space. While applying the trapezoidal decomposition we add nodes to both in the centers of trapezoids and vertical walls. So called a robot want to traverse between obstacles it avoids from this obstacle with following the nodes added. That nodes create a graph which provide smooth traverse between the obstacles at least we assume that way. In this case we assume that we are creating a graph with using all nodes. Here is my question, Is it possible to plan a graph with only using the center of trapezoid or the center nodes of vertical walls? My idea is; it depends on the situation(the position of obstacles, the shape of the robot etc.) But I could not image such graph which could be changed to a new graph with only using the nodes on the vertical walls. If we design the example do we need to increase the number of edges? I think it is not necessity to increase the number of edges to manage to use only vertical wall’s nodes but I could not figure out how it could be possible exactly.