## Interval representations of planar graphs.(English)Zbl 0595.05027

A graph is said to be a strict d-box graph if it is an intersection graph such that the sets are closed d-boxes $$(=d$$-dimensional closed intervals) in $$R^ d$$, no two of which have an interior point in common and such that two boxes which intersect have precisely a (d-1)-box in common. The author proves that every planar graph is a strict 3-box graph, and characterizes the 2-box graphs; these turn out to be precisely the proper subgraphs of 4-connected planar triangulations. In the case when more than one rectangle is allowed to represent a vertex, the author shows that two rectangles for each vertex are sufficient to represent any planar graph $$(d=2)$$.
Reviewer: J.Širáň

### MSC:

 05C10 Planar graphs; geometric and topological aspects of graph theory
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### References:

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