### abstract

- This paper considers the problem of distributively constructing a minimum-weight spanning tree (MST) for graphs of constant diameter in the bounded-messages model, where each message can contain at most B bits for some parameter B. It is shown that the number of communication rounds necessary to compute an MST for graphs of diameter 4 or 3 can be as high as \(\Omega(\sqrt[3]n/\sqrt{B})\) and \(\Omega(\sqrt[4]n/\sqrt{B})\), respectively. The asymptotic lower bounds hold for randomized algorithms as well. On the other hand, we observe that O(log n) communication rounds always suffice to compute an MST deterministically for graphs with diameter 2, when B = O(log n). These results complement a previously known lower bound of \(\Omega(\sqrt[2]n/B)\) for graphs of diameter Ω(log n).