Some properties of edge intersection graphs of single bend paths on a grid

We consider the family of intersection graphs G of paths on a grid, where every vertex v in G corresponds to a single bend path Pv on a grid, and two vertices are adjacent in G if and only if the corresponding paths share an edge on the grid. We first show that these graphs have the Erdِs-Hajnal property. Then we present some properties concerning the neighborhood of a vertex in these graphs, and finally we consider some subclasses of chordal graphs for which we give necessary and sufficient conditions to be edge intersection graphs of single bend paths in a grid.