Submatrices of non-tree-realizable distance matrices

We deal with distance matrices of real (this means, not necessarily integer) numbers. It is known that a distance matrix D of order n is tree-realizable if and only if all its principal submatrices of order 4 are tree-realizable. We discuss bounds for the number, denoted Q_i(D), of non-tree-realizable principal submatrices of order i ? 4 of a non-tree-realizable distance matrix D of order n?i, and we construct some distance matrices which meet extremal conditions on Q_i(D). Our starting point is a proof that a non-tree-realizable distance matrix of order 5 has at least two non-tree-realizable principal submatrices of order 4. Optimal realizations (by graphs with circuits) of distance matrices which are not tree-realizable are not yet as well known as optimal realizations which are trees. Using as starting point the optimal realization of the (arbitrary) distance matrix of order 4, we investigate optimal realizations of non-tree-realizable distance matrices with the minimum number of non-tree-realizable principal submatrices of order 4.