Bounded Tolerance Representations for Maximal Outerplanar Graphs

Abstract

A graph G = (V, E) is a tolerance graph if there is a set I = {Iv∣v ε V} of closed real interval and a set τ = {tv∣v ε V} of positive real numbers such that (x, y) ε E ⇔ ∣Ix ∩ Iy∣ ≥ min{τx, τy}. We show that if G is a 2-connected maximal outerplanar graph with more than two vertices of degree 2, then G has S3 as an induced subgraph. We provide a characterization of the class of 2-connected maximal outerplanar graphs that are bounded tolerance graphs.