Consider a simple graph $G$ where each edge is either red or blue. I'm interested in the following notion of connectivity:
Two vertices $u$ and $v$ are said to be connected if there is a path connecting them consisting only of red edges and another consisting only of blue edges.
I'm particularly interested in what I think of as "totally disconnected graphs", that is 2-edge-colored graphs such that every induced subgraph is disconnected (meaning that there are two vertices which are not connected according to the above definition of connectivity).
Have these notions been considered in the literature?