Edge intersection graphs of single bend paths on a grid

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 B ₁-EPG representations and the corresponding graphs are called B ₁-EPG graphs. We prove that any tree is a B₁-EPG graph. Moreover, we give a structural property that enables one to generate non B ₁-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.