For an irreducible stochastic matrix

$T$, the author considers a certain condition number

$c\left(T\right)$, which measures the stability of the corresponding stationary distribution when

$T$ is perturbed, and then characterizes the strongly connected directed graphs

$D$ such that

$c\left(T\right)$ is bounded as

$T$ ranges over

${\mathcal{S}}_{D}$, the set of stochastic matrices whose directed graph is contained in

$D$. For those digraphs

$D$ for which

$c\left(T\right)$ is bounded, the maximum value of

$\left\{c\right(T)\mid T$ ranges over

${\mathcal{S}}_{D}\}$ is determined.