The complexity of 2-intersection graphs of 3-hypergraphs recognition for claw-free graphs and triangulated claw-free graphs

Given a 3-uniform hypergraph H, its 2-intersection graph G has as vertex set the hyperedges of H and ee′ is an edge of G whenever e and e′ have exactly two common vertices in H. Di Marco et al. prove in Di Marco et al. (2023) that deciding whether a graph G is the 2-intersection graph of a 3-uniform hypergraph is NP-complete. Following this result, we study the class of claw-free graphs. We show that the recognition problem remains NP-complete for that class, but becomes polynomial if we consider triangulated claw-free graphs.