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

Niccolò Di Marco, Andrea Frosini, Christophe Picouleau

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