Edge intersection graphs of single bend paths on a grid

Abstract

AbstractWe 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