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

Abstract

Edge-intersection graphs of paths on a grid (or EPG graphs) are graphs whose vertices can be represented as simple paths on a rectangular grid such that two vertices are adjacent in the graph if and only if the corresponding paths share at least one edge of the grid. For two boundary points p and q on two adjacent boundaries of a rectangular grid G, we call the unique single-bend path connecting p and q in G using no other boundary point of G as the path generated by (p,q). A path in G is called boundary-generated, if it is generated by some pair of points on two adjacent boundaries of G. In this article, we study the edge-intersection graphs of boundary-generated paths on a grid or ∂EPG graphs. The motivation for studying these graphs comes from problems in the context of circuit layout.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.