Edge intersection graphs of L-shaped paths in grids

Abstract

a b s t r a c tIn 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: , , , , and we consider zero bend paths (i.e., | and -) to be degenerate '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., { }, { , }, { , }, and { , , }) and we denote the classes by [ ], [ , ], [ , ], and [ , , ] 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., [ ] [ , ], [ , ] [ , , ] B 1 -EPG and [ , ] is incomparable with [ , ]). Additionally, we give characterizations and polytime recognition algorithms for special subclasses of Split ∩ [ ].

On Edge Intersection Graphs of Paths with 2 Bends

“…There are two main branches in this kind of research. The first one is understanding the structure of graphs with at most k bends -so far, the case of 1-bend graphs received most attention [12,8,10,2]. The other is finding the smallest k, such that every graph of a given class G is a k-bend graph.…”

“…Booth and Lueker [5]). The recognition of 1-bend graphs is NP-complete (see Heldt et al [13]), even if the representation is restricted to any prescribed set of 1-bend objects (see Cameron et al [8]). However, the problem becomes trivially solvable when k is at least the maximum degree of the input graph [13].…”

“…Namely, we show that it is NP-complete to recognize 2-bend graphs. We also consider some restrictions, where we permit just some types of the curves in an EPG-representation (similarly to [8]). One of these restrictions, i.e., monotonic EPG-representations, where each path ascends in rows and columns, was already considered by Golumbic et al [12].…”

“…After this generalization, we may ask about particular restrictions. These restrictions are represented by restricting number of slopes that segments may use or even by using just prescribed shapes (in a flavor similar to [8]).…”

“…For unaligned EPG-graphs, we show that it is NP-hard to determine whether a graph is an unaligned 2-bend graph (hardness of the recognition for 1-bend graphs follows from [8]).…”