On bandwidth of graphs I am trying to find references on algorithms for graphs of bounded bandwidth, in the same way as it is done with treewidth for instance.
I could only find research related to  computing the bandwidth, or properties of this measure, but not using it as assumption for better algorithms.

Also, I am very interested in a generalization of bandwidth in higher dimensions. For instance this paper studies 2-dimensional bandwidth, but considering only the $L^1$ and $L^\infty$ norms on $\mathbb N\times \mathbb N$, whereas I am more interested in the euclidean norm $L^2$. It seems natural to consider graphs of $n$-dimensional bandwidth for euclidean norm. Formally, the $n$-dimensional bandwidth of a graph $G=(V,E)$ is defined as :
$$\min_{\alpha} ~\max_{(x,y)\in E}||\alpha(x)-\alpha(y)||_2$$
where $\alpha$ ranges over injective functions $V\to \mathbb N^n$.

This is pretty natural, for instance graphs coming from discretizations of real-life systems following differential equations would likely have bounded bandwidth. Indeed, if an edge takes you to the state of the system at the next time instant, it can not be too far from your current state if the time step is small enough and the system evolves continuously. It seems that this special structure could be used to design better algorithms (in particular for solving games on these graphs), but I could not find anything on this kind 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".