A Necessary Condition and a Sufficient Condition for Pairwise Compatibility Graphs

In this paper we give a necessary condition and a sufficient condition for a graph to be a pairwise compatibility graph (PCG). Let G be a graph and let Gc be the complement of G. We show that if Gc has two disjoint chordless cycles then G is not a PCG. On the other hand, if Gc has no cycle then G is a PCG. Our conditions are the first necessary condition and the first sufficient condition for pairwise compatibility graphs in general. We also show that there exist some graphs in the gap of the two conditions which are not PCGs. Submitted: May 2016 Reviewed: November 2016 Revised: December 2016 Accepted: December 2016 Final: January 2017 Published: February 2017 Article type: Regular paper Communicated by: M. Kaykobad and R. Petreschi Work of Debajyoti Mondal is partially supported by NSERC. E-mail addresses: [email protected] (Md. Iqbal Hossain) [email protected] (Sammi Abida Salma) [email protected] (Md. Saidur Rahman) [email protected] (Debajyoti Mondal) 342 Hossain et al. Conditions for Pairwise Compatibility Graphs