  A graph $G$ is a $B_0$-VPG graph if one can associate a path on a rectangular
grid with each vertex such that two vertices are adjacent if and only if the
corresponding paths intersect at at least one grid-point. A graph $G$ is a
contact $B_0$-VPG graph if it is a $B_0$-VPG graph admitting a representation
with no two paths crossing and no two paths sharing an edge of the grid. In
this paper, we present a minimal forbidden induced subgraph characterisation of
contact $B_0$-VPG graphs within four special graph classes: chordal graphs,
tree-cographs, $P_4$-tidy graphs and $P_5$-free graphs. Moreover, we present a
polynomial-time algorithm for recognising chordal contact $B_0$-VPG graphs.
