Edge Intersection Graphs of L-Shaped Paths in Grids

In this paper we continue the study of the edge intersection graphs of single bend paths on a rectangular grid (i.e., the edge intersection graphs where each vertex is represented by one of the following shapes: x, p, y, q). These graphs, called B1-EPG graphs, were first introduced by Golumbic et al (2009) [5]. We consider the natural subclasses of B1-EPG formed by the subsets of the four single bend shapes (i.e., {x}, {x, p}, {x, q}, and {x, p, q}) and we denote the classes by [x], [x, p], [x, q], and [x, p, q] respectively note: all other subsets are isomorphic to these up to 90 degree rotation. We show that testing for membership in each of these classes is NP-complete and observe the expected strict inclusions and incomparability (i.e., [x] ( [x, p], [x, q] ( [x, p, q] ( B1-EPG and [x, p] is incomparable with [x, q]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split ∩ [x].