Bounded Tolerance Representations for Maximal Outerplanar Graphs

Abstract

A graph G = (V, E) is a tolerance graph if there is a set I = {I(subscript υ)|υ ∈ V} of closed real interval and a set т = { т(subscript υ)|υ ∈ V} of positive real numbers such that (x, y) ∈ E ↔ |I(subscript x)∩I(subscript y)| ≥ min{ т(subscript x), т(subscript y)}. We show that if G is a 2-connected maximal outer-planar 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 outer-planar graphs that are bounded tolerance graphs.