Edge-intersection graphs of boundary-generated paths in a grid

We show that ∂EPG graphs can be covered by two collections of vertex-disjoint co-bipartite chain graphs. This leads us to a linear-time testable characterization of ∂EPG trees and also an almost tight upper bound on the equivalence covering number of general ∂EPG graphs. We also study the cases of two-sided ∂EPG and three-sided ∂EPG graphs, which are respectively, the subclasses of ∂EPG graphs obtained when all the boundary-vertex pairs which generate the paths are restricted to lie on at most two or three boundaries of the grid. For the former case, we give a complete characterization.