  A universal representation theorem is derived that shows any graph is the
intersection graph of one chordal graph, a number of co-bipartite graphs, and
one unit interval graph. Central to the the result is the notion of the clique
cover width which is a generalization of the bandwidth parameter. Specifically,
we show that any planar graph is the intersection graph of one chordal graph,
four co-bipartite graphs, and one unit interval graph. Equivalently, any planar
graph is the intersection graph of a chordal graph and a graph that has {clique
cover width} of at most seven. We further describe the extensions of the
results to graphs drawn on surfaces and graphs excluding a minor of crossing
number of at most one.
