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This paper applies a simple constrained ordering for the solution of the equality constrained state estimation problem. By low-rank perturbations in the semidefinite (1,1) block of the coefficient matrix, while maintaining sparsity, a saddle point matrix is formed. The vectors used for generating the perturbations are rows of the matrix associated with the equality constraints that represent the zero injections. The proposed algorithm make use of the Bridson's ordering constraint for saddle-point systems, which is sufficient to guarantee the existence of a signed Cholesky factorization for the perturbed indefinite coefficient matrix, with separate symbolic and numerical phases. The need for numerical pivoting during factorization is avoided, with clear benefits for performance. Two alternative implementations are provided, either modifying a fill-reducing ordering algorithm to incorporate this constraint or modifying an existing fill-reducing ordering to respect the constraint. The proposed method is compared with existing methods in terms of computational time and convergence robustness. The IEEE 300-bus and the FRCC 3949-bus systems are used as test beds for this study.