Helly $\mathbf{EPT}$ graphs on bounded degree trees: forbidden induced subgraphs and efficient recognition

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 ∈ 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. In [6], Golumbic et al. wonder if the only obstructions for an EPT graph belonging to [h, 2, 2] are the chordless cycles Cn 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 ≥ 4 inside the class EPT . As a byproduct, we prove that Helly EPT∩[h, 2, 2] = Helly [h, 2, 2]. Following the approach used in [10], 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.