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: ⌞ , ⌜ , ⌟ , ⌝ ). These graphs, called B 1 - EPG graphs, were first introduced by Golumbic et al (2009) [Golumbic, M. C., M. Lipshteyn and M. Stern, Edge intersection graphs of single bend paths on a grid, Networks 54:3 (2009), 130–138]. We focus on the class [⌞] (the edge intersection graphs of ⌞-shapes) and show that testing for membership in [⌞] is NP-complete. We then give a characterization and polytime recognition algorithm for special subclasses of Split ∩ [ ⌞ ] . We also consider the natural subclasses of B 1 -EPG formed by the subsets of the four single bend shapes (i.e., { ⌞ } , { ⌞ , ⌜ } , { ⌞ , ⌝ } , { ⌞ , ⌜ , ⌝ } – note: all other subsets are isomorphic to these up to 90 degree rotation). We observe the expected strict inclusions and incomparability (i.e., [ ⌞ ] ⊊ [ ⌞ , ⌜ ] , [ ⌞ , ⌝ ] ⊊ [ ⌞ , ⌜ , ⌝ ] ⊊ B 1 -EPG and [ ⌞ , ⌜ ] is incomparable with [ ⌞ , ⌝ ] ).