Biclique graphs of split graphs

• Published in: Discrete Applied Mathematics Vol. 323; pp. 252 - 267
• Main Authors: Groshaus, M., Guedes, A.L.P., Puppo, J.P.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 31-12-2022
• Subjects: [formula omitted]-complete; Biclique graphs; Biclique-complete; Bicliques; Split graphs; Split graphs; Biclique-complete; Bicliques; NP-complete; Biclique graphs
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2021.12.028

The biclique graph is the intersection graph of the bicliques of a graph. Its recognition problem is still open. In this paper we study the problem restricted to the class of split graphs. We define a class called nested separable split graphs and solve the recognition problem of biclique graphs of a subclass of the mentioned class. For that, we give a polynomial time algorithm with positive certificate. We also characterize nested split graphs and study properties of biclique graphs of split graphs. For example, we show that they are Hamiltonian. We also present a complexity result for recognizing graphs such that their biclique graphs are complete graphs, that is, we prove that the problem of recognizing biclique-complete graphs in co-NP-complete.