Edge intersection graphs of L-shaped paths in grids

In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes left perpendicular inverted right perpendicular, right perpendicular inverted left perpendicular, and we consider zero bend paths (i.e., vertical bar and -) to be degenerate left perpendicular 's. These graphs, called B-1-EPG graphs, were first introduced by Golumbic et al. (2009). We consider the natural subclasses of B-1-EPG formed by the subsets of the four single bend shapes (i.e., {left perpendicular }, {left perpendicular, inverted right perpendicular }, {left perpendicular , inverted left perpendicular}, and {left perpendicular , inverted right perpendicular, inverted left perpendicular}) and we denote the classes by [left perpendicular], [left perpendicular, inverted right perpendicular], [left perpendicular , inverted left perpendicular], and [left perpendicular , inverted right perpendicular, inverted left perpendicular] 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., [left perpendicular] not subset of [left perpendicular, inverted right perpendicular], [left perpendicular, inverted left perpendicular] not subset of [left perpendicular, inverted right perpendicular, inverted left perpendicular] not subset of B-1-EPG and [left perpendicular, inverted right perpendicular] is incomparable with [left perpendicular, inverted left perpendicular] ). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split boolean AND[left perpendicular]. (C) 2015 Elsevier B.V. All rights reserved.