Geodesic
A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 1, pp. 13-28

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We characterize the class L_3^2 of intersection graphs of hypergraphs with rank at most 3 and multiplicity at most 2 by means of a finite list of forbidden induced subgraphs in the class of threshold graphs. We also give an O(n)-time algorithm for the recognition of graphs from L_3^2 in the class of threshold graphs, where n is the number of vertices of a tested graph.
Keywords: intersection graph, hypergraph rank, hypergraph multiplicity, forbidden induced subgraph, threshold graph 
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Metelsky, Yury; Schemeleva, Kseniya; Werner, Frank. A Finite Characterization and Recognition of Intersection Graphs of Hypergraphs with Rank at Most 3 and Multiplicity at Most 2 in the Class of Threshold Graphs. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 1, pp. 13-28. http://geodesic.mathdoc.fr/item/DMGT_2017_37_1_a1/