  In this paper the problem of finding the sparsest (i.e., minimum cardinality)
critical $k$-tuple including one arbitrarily specified measurement is
considered. The solution to this problem can be used to identify weak points in
the measurement set, or aid the placement of new meters. The critical $k$-tuple
problem is a combinatorial generalization of the critical measurement
calculation problem. Using topological network observability results, this
paper proposes an efficient and accurate approximate solution procedure for the
considered problem based on solving a minimum-cut (Min-Cut) problem and
enumerating all its optimal solutions. It is also shown that the sparsest
critical $k$-tuple problem can be formulated as a mixed integer linear
programming (MILP) problem. This MILP problem can be solved exactly using
available solvers such as CPLEX and Gurobi. A detailed numerical study is
presented to evaluate the efficiency and the accuracy of the proposed Min-Cut
and MILP calculations.
