  The problem of characterizing maximal non-Hamiltonian graphs may be naturally
extended to characterizing graphs that are maximal with respect to
non-traceability and beyond that to $t$-path traceability. We show how
traceability behaves with respect to disjoint union of graphs and the join with
a complete graph. Our main result is a decomposition theorem that reduces the
problem of characterizing maximal $t$-path traceable graphs to characterizing
those that have no universal vertex. We generalize a construction of maximal
non-traceable graphs by Zelinka to $t$-path traceable graphs.
