Deterministic graph connectivity in the broadcast congested clique

We present a deterministic constant-round protocol for the graph connectivity problem in the model where each of the $n$ nodes of a graph receives a row of the adjacency matrix, and broadcasts a single sublinear size message to all other nodes. Communication rounds are synchronous. This model is sometimes called the broadcast congested clique. Specifically, we exhibit a deterministic protocol that computes the connected components of the input graph in three rounds, each player communicating $\mathcal{O}(\sqrt{n} \cdot \log n)$ bits per round. Our result is based on a $d$-pruning protocol, which consists in successively removing nodes of degree at most $d$ until obtaining a graph with minimum degree larger than $d$.