Abstract
We prove that every planar graph is the intersection graph of a collection of three-dimensional boxes, with intersections occuring only in the boundaries of the boxes. Furthermore, we characterize the graphs that have such representations (called strict representations) in the plane. These are precisely the proper subgraphs of 4-connected planar triangulations, which we characterize by forbidden sub-graphs. Finally, we strengthen a result of E. R. Scheinerman (“Intersection Classes and Multiple Intersection Parameters”, Ph. D. thesis, Princeton Univ., 1984) to show that every planar graph has a strict representation using at most two rectangles per vertex.
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