  An undirected graph G is called a VPT graph if it is the vertex intersection
graph of a family of paths in a tree. The class of graphs which admit a VPT
representation in a host tree with maximum degree at most h is denoted by
[h,2,1]. The classes [h,2,1] are closed by taking induced subgraphs, therefore
each one can be characterized by a family of minimal forbidden induced
subgraphs. In this paper we associate the minimal forbidden induced subgraphs
for [h,2,1] which are VPT with (color) h-critical graphs. We describe how to
obtain minimal forbidden induced subgraphs from critical graphs, even more, we
show that the family of graphs obtained using our procedure is exactly the
family of VPT minimal forbidden induced subgraphs for [h,2,1]. The members of
this family together with the minimal forbidden induced subgraphs for VPT, are
the minimal forbidden induced subgraphs for [h,2,1], with $h\geq 3$. Notice
that by taking h=3 we obtain a characterization by minimal forbidden induced
subgraphs of the class VPT $\cap$ EPT=EPT $\cap$ Chordal=[3,2,2]=[3,2,1].
