In this paper we consider the Maximum Independent Set problem (MIS) on @math -EPG graphs. EPG (for Edge intersection graphs of Paths on a Grid) was introduced in edgeintersinglebend as the class of graphs whose vertices can be represented as simple paths on a rectangular grid so that two vertices are adjacent if and only if the corresponding paths share at least one edge of the underlying grid. The restricted class @math -EPG denotes EPG-graphs where every path has at most @math bends. The study of MIS on @math -EPG graphs has been initiated in wadsMIS where authors prove that MIS is NP-complete on @math -EPG graphs, and provide a polynomial @math -approximation. In this article we study the approximability and the fixed parameter tractability of MIS on @math -EPG. We show that there is no PTAS for MIS on @math -EPG unless P @math NP, even if there is only one shape of path, and even if each path has its vertical part or its horizontal part of length at most @math . This is optimal, as we show that if all paths have their horizontal part bounded by a constant, then MIS admits a PTAS. Finally, we show that MIS is FPT in the standard parameterization on @math -EPG restricted to only three shapes of path, and @math -hard on @math -EPG. The status for general @math -EPG (with the four shapes) is left open.
We combine the known notion of the edge intersection graphs of paths in a tree with a VLSI grid layout model to introduce the edge intersection graphs of paths on a grid. Let P be a collection of nontrivial simple paths on a grid G. We define the edge intersection graph EPG(P) of P to have vertices which correspond to the members of P, such that two vertices are adjacent in EPG(P) if the corresponding paths in P share an edge in G. An undirected graph G is called an edge intersection graph of paths on a grid (EPG) if G = EPG(P) for some P and G, and 〈P,G〉 is an EPG representation of G. We prove that every graph is an EPG graph. A turn of a path at a grid point is called a bend. We consider here EPG representations in which every path has at most a single bend, called B1-EPG representations and the corresponding graphs are called B1-EPG graphs. We prove that any tree is a B1-EPG graph. Moreover, we give a structural property that enables one to generate non B1-EPG graphs. Furthermore, we characterize the representation of cliques and chordless 4-cycles in B1-EPG graphs. We also prove that single bend paths on a grid have Strong Helly number 3. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009 Abstract 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 (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 [ ⌞ , ⌝ ] ). The bend-numberb(G) of a graph G is the minimum k such that G may be represented as the edge intersection graph of a set of grid paths with at most k bends. We confirm a conjecture of Biedl and Stern showing that the maximum bend-number of outerplanar graphs is 2. Moreover we improve the formerly known lower and upper bound for the maximum bend-number of planar graphs from 2 and 5 to 3 and 4, respectively.