Minimum Spanning Tree in Deterministic Linear Time For Graphs of High Girth

We show a simple deterministic linear time algorithm for computing the minimum spanning tree of graphs on n vertices with girth at least b(t, lg n), where t ≥ 1 is a constant and b(x, y) is a variant of the inverse of the fast-growing Ackermann function. We also prove: (i) A deterministic linear time algorithm for general graphs follows from an algorithm for graphs with girth at least g for any constant g ≥ 1; (ii) To find the minimum spanning tree of a graph G, it suffices to find the minimum spanning of any graph Ĝ obtained from G by removal of O(n/b(t, lg n)) edges for any constant t ≥ 1; (iii) Our high girth algorithm yields an alternative deterministic linear time algorithm for random graphs.