Gridline graphs: a review in two dimensions and an extension to higher dimensions Original Research Article

Gridline graphs can be realized in the plane with vertices adjacent whenever they are on a common vertical or horizontal line. We review some applications and characterizations, e.g. they are line graphs of bipartite graphs, and provide practical O(|V|3) algorithms for some classical problems. We extend gridline graphs from the plane to higher dimensions. We characterize these graphs using a vertex coloring of the clique graph that corresponds to the conditions that, in the intersection graph of lines parallel to axes in Rn, any cycle of four lines must remain in two dimensions and any path between two given lines must traverse the same two dimensions.