  We consider a stochastic directed graph on the integers whereby a directed
edge between $i$ and a larger integer $j$ exists with probability $p_{j-i}$
depending solely on the distance between the two integers. Under broad
conditions, we identify a regenerative structure that enables us to prove limit
theorems for the maximal path length in a long chunk of the graph. The model is
an extension of a special case of graphs studied by Foss and Konstantopoulos,
Markov Process and Related Fields, 9, 413-468. We then consider a similar type
of graph but on the `slab' $\Z \times I$, where $I$ is a finite partially
ordered set. We extend the techniques introduced in the in the first part of
the paper to obtain a central limit theorem for the longest path. When $I$ is
linearly ordered, the limiting distribution can be seen to be that of the
largest eigenvalue of a $|I| \times |I|$ random matrix in the Gaussian unitary
ensemble (GUE).
