  In this paper we investigate the $ L^1 $-Liouville property, underlining its
connection with stochastic completeness and other structural features of the
graph. We give a characterization of the $ L^1 $-Liouville property in terms of
the Green function of the graph and use it to prove its equivalence with
stochastic completeness on model graphs. Moreover, we show that there exist
stochastically incomplete graphs which satisfy the $ L^1 $-Liouville property
and prove some comparison theorems for general graphs based on inner-outer
curvatures. We also introduce the Dirichlet $L^1$-Liouville property of
subgraphs and prove that if a graph has a Dirichlet $L^1$-Liouville subgraph,
then it is $L^1$-Liouville itself. As a consequence, we obtain that the $
L^1$-Liouville property is not affected by a finite perturbation of the graph
and, just as in the continuous setting, a graph is $ L^1$-Liouville provided
that at least one of its ends is Dirichlet $ L^1$-Liouville.
