Intersection Dimensions of Graph Classes

The intersection dimension of a graphG with respect to a classA of graphs is the minimumk such thatG is the intersection of at mostk graphs on vertex setV(G) each of which belongs toA. We consider the question when the intersection dimension of a certain family of graphs is bounded or unbounded. Our main results are (1) ifA is hereditary, i.e., closed on induced subgraphs, then the intersection dimension of all graphs with respect toA is unbounded, and (2) the intersection dimension of planar graphs with respect to the class of permutation graphs is bounded. We also give a simple argument based on Ben-Arroyo Hartman, I., Newman, I., Ziv, R.:On grid intersection graphs, Discrete Math.87 (1991) 41-52] why the boxicity (i.e., the intersection dimension with respect to the class of interval graphs) of planar graphs is bounded. Further we study the relationships between intersection dimensions with respect to different classes of graphs.