On bandwidth of graphs
As you mentioned that "My main goal is to solve different kinds of games on these graphs, but I'm curious about other problems too", you can have a look at the thesis by Morgan Chopin -- "Optimization problems with propagation in graphs: Parameterized complexity and approximation" (https://tel.archives-ouvertes.fr/tel-00933769). 
Here, author shows some results with "bandwidth" for "Firefighter" problem and "Target Set Selection" problem which can be thought of as a discretized version of "Influence Maximization".