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: $\llcorner$,$\ulcorner$, $\urcorner$, $\lrcorner$, and we consider zero bend paths (i.e., | and $-$) to be degenerate $\llcorner$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., {$\llcorner$}, {$\llcorner$,$\ulcorner$}, {$\llcorner$,$\urcorner$}, and {$\llcorner$,$\ulcorner$,$\urcorner$}) and we denote the classes by [$\llcorner$], [$\llcorner$,$\ulcorner$], [$\llcorner$,$\urcorner$], and [$\llcorner$,$\ulcorner$,$\urcorner$] 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., [$\llcorner$] $\subsetneq$ [$\llcorner$,$\ulcorner$], [$\llcorner$,$\urcorner$] $\subsetneq$ [$\llcorner$,$\ulcorner$,$\urcorner$] $\subsetneq$ $B_1$-EPG; also, [$\llcorner$,$\ulcorner$] is incomparable with [$\llcorner$,$\urcorner$]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split $\cap$ [$\llcorner$].