  The edge intersection graph of a family of paths in host tree is called an
$EPT$ graph. When the host tree has maximum degree $h$, we say that $G$ belongs
to the class $[h,2,2]$. If, in addition, the family of paths satisfies the
Helly property, then $G \in$ Helly $[h,2,2]$.
  The time complexity of the recognition of the classes
  $[h,2,2]$ inside the class $EPT$ is open for every $h> 4$. Golumbic et al.
wonder if the only obstructions for an $EPT$ graph belonging to $[h,2,2]$ are
the chordless cycles $C_n$ for $n> h$. In the present paper, we give a negative
answer to that question, we present a family of $EPT$ graphs which are
forbidden induced subgraphs for the classes $[h,2,2]$.
  Using them we obtain a total characterization by induced forbidden subgraphs
of the classes Helly $[h,2,2]$ for $h\geq 4$ inside the class $EPT$. As a
byproduct, we prove that Helly $EPT$$\cap [h,2,2]=$ Helly $[h,2,2]$. We
characterize Helly $[h,2,2]$ graphs by their atoms in the decomposition by
clique separators. We give an efficient algorithm to recognize Helly $[h,2,2]$
graphs.

Combinatorics;Discrete Mathematics