I believe Herbert Edelsbrunner's work on alpha shapes may have some relevance, though when I was reading about it I was mainly interested in applications to determining volume and surface areas so I'm not sure if the precise question you are asking is addressed. See the papers here on his website, perhaps starting with the paper: H. Edelsbrunner. The union of balls and its dual shape. Discrete Comput. Geom. 13 (1995), 415-440.

I believe Herbert Edelsbrunner's work on alpha shapes may have some relevance, though when I was reading about it I was mainly interested in applications to determining volume and surface areas so I'm not sure if the precise question you are asking is addressed. See the papers here on his website, perhaps starting with the paper: H. Edelsbrunner. The union of balls and its dual shape. Discrete Comput. Geom. 13 (1995), 415-440.

The abstract of that paper:

Efficient algorithms are described for computing topological, combinatorial, and metric properties of the union of finitely many spherical balls in $\mathbb{R}^d$ . These algorithms are based on a simplicial complex dual to a decomposition of the union of balls using Voronoi cells, and on short inclusion-exclusion formulas derived from this complex. The algorithms are most relevant in $\mathbb{R}^3$ where unions of finitely many balls are commonly used as models of molecules.

I think the main idea is basically that suggested by Anton Petrunin.