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Study proves optimal separators exist for intersection graphs of spheres, indicating vital structural properties.
We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension d. One of our results is that if an intersection graph of n spheres in ℝ^d has m edges, then it contains a balanced separator of size O_d(m1/dn1-2/d). This bound is best possible in terms of the parameters involved. The same result holds if the balls and spheres are replaced by fat convex bodies and their boundaries.
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Fox et al. (2026) studied this question.
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