Edge intersection graphs of paths on a grid

A graph G is an edge intersection graph of paths on a grid (or EPG graph) if its vertices can be represented as simple paths on a rectangular grid, such that two vertices are adjacent in the graph, if and only if their corresponding paths share at least one edge of the grid. EPG graphs were first introduced and studied by M. C. Golumbic, M. Lipshteyn, and M. Stern. In that paper, the authors show that every graph is an EPG graph. Moreover, it always has a monotonic EPG representation, namely, one where each path is ascending in rows and columns. A turn of a path at a grid-point is called a bend, and a graph is called a k-bend EPG graph (denoted Bk-EPG), if it has an EPG representation in which each path has at most k bends. A graph G is chordal if G does not contain a chordless cycle of size at least four, often called a hole.