The approach outlined in the following introduces a new way to simulate the physics of 2D game characters that cannot jump. The focus of the paper is on exploring the possibility of representing characters as a set of two points on a graph. These points move along the edges and resolve collisions. This was tested by implementing a prototype and developing several features around it. While the prototype was functional for a multitude cases, there were several exceptions that could not be solved with the current set of rules. A graph-based method was able to move the characters and to resolve collisions, but it failed at handling all cases that needed characters to be stacked on-top of each other. A graph-based approach is problematic when characters have a width, collide with each other and need to fall down ledges. If a game requires all three of these, it is recommended to use traditional collisions instead of the graph-based approach. If the game does not e.g. require the characters to collide with each other a graph-based solution might be beneficial.