Triangle-Free Outerplanar 3-Graphs Are Pairwise Compatibility Graphs

A graph G = (V,E) is called a pairwise compatibility graph (PCG) if there exists an edge-weighted tree T and two non-negative real numbers dmin and dmax such that each vertex u ′ ∈ V corresponds to a leaf u of T and there is an edge (u′, v′) ∈ E if and only if dmin ≤ dT (u, v) ≤ dmax in T . Here, dT (u, v) denotes the distance between u and v in T , which is the sum of the weights of the edges on the path from u to v. It is known that not all graphs are PCGs. Thus it is interesting to know which classes of graphs are PCGs. In this paper we show that triangle-free outerplanar graphs with the maximum degree 3 are PCGs. Submitted: April 2012 Reviewed: July 2012 Revised: December 2012 Accepted: December 2012 Final: December 2012 Published: January 2013 Article type: Regular paper Communicated by: S.-i. Nakano E-mail addresses: [email protected] (Sammi Abida Salma) [email protected] (Md. Saidur Rahman) [email protected] (Md. Iqbal Hossain) 82 Salma, Rahman, Hossain Triangle-Free Outerplanar 3-Graphs are PCGs