  Given a stochastic matrix $P$ partitioned in four blocks $P_{ij}$, $i,j=1,2$,
Kemeny's constant $\kappa(P)$ is expressed in terms of Kemeny's constants of
the stochastic complements $P_1=P_{11}+P_{12}(I-P_{22})^{-1}P_{21}$, and
$P_2=P_{22}+P_{21}(I-P_{11})^{-1}P_{12}$. Specific cases concerning periodic
Markov chains and Kronecker products of stochastic matrices are investigated.
Bounds to Kemeny's constant of perturbed matrices are given. Relying on these
theoretical results, a divide-and-conquer algorithm for the efficient
computation of Kemeny's constant of graphs is designed. Numerical experiments
performed on real-world problems show the high efficiency and reliability of
this algorithm.
