Inverse M-matrix completions of patterns omitting some diagonal positions Original Research Article

Let N={1,…,n}. Which subsets Q of N×N have the property that whenever aij for (i,j)set membership, variantQ form a “partial inverse M-matrix” (i.e., aijgreater-or-equal, slanted0 and if L×Lsubset of or equal toQ, then the inverse of {aij:i,jset membership, variantL} is an M-matrix), aij can be defined for (i,j)negated set membershipQ so that the inverse of A=[aij] is an M-matrix? In terms of a digraph G with vertex set N and arcs (i,j)set membership, variantQwith i≠j, our results say (1) it is necessary that the induced subdigraph of any alternate path to an arc be a clique, (2) a cycle Γ need not be contained in a clique if (i,i)negated set membershipQ for at least one vertex i of Γ, (3) when n=4, it is necessary and sufficient that condition (1) is true and every simple cycle has a vertex i with (i,i)negated set membershipQ, and (4) the general question can be reduced to the case in which G is strongly connected, much as it has been when Q contains (i,i) for every iset membership, variantN.