Sign determinacy of M-matrix minors

We show that if A is an M-matrix for which the length of the longest simple cycle in its associated undirected graph G(A) is at most 3, then every minor of A has determined sign (nonnegative or nonpositive), independent of the magnitudes of the matrix entries. Consequently, if A and B are M-matrices such that G(A) and G(B) are subgraphs of an undirected graph with longest simple cycle at most 3, then all principal minors of AB are nonnegative.