  A bipartite graph is chordal bipartite if every cycle of length at least 6
has a chord in it. In this paper, we investigate the structure of $P_5$-free
chordal bipartite graphs and show that these graphs have a Nested Neighborhood
Ordering, a special ordering among its vertices. Further, using this ordering,
we present polynomial-time algorithms for classical problems such as
Hamiltonian cycle (path) and longest path. Two variants of Hamiltonian path
include Steiner path and minimum leaf spanning tree, and we obtain
polynomial-time algorithms for these problems as well restricted to $P_5$-free
chordal bipartite graphs.
