Sliding column model for t-unit bar visibility representations of graphs

Abstract

A bar visibility representation of a graph [Formula: see text] is an assignment of the vertices of [Formula: see text] to distinct horizontal line segments in the plane so that two vertices are adjacent in [Formula: see text] if and only if there is an uninterrupted vertical channel of positive width that joins the bars corresponding to those vertices. A bar visibility representation is called a unit bar visibility representation if every bar has the same length. If each vertex is assigned to [Formula: see text] distinct bars of the same length in a unit bar visibility representation then the representation is called [Formula: see text]-unit bar visibility representation. In this paper, we introduce a “sliding column model” for [Formula: see text]-unit bar visibility representation and show that every graph of maximum degree [Formula: see text] has a [Formula: see text]-unit bar visibility representation for [Formula: see text]. We also show that a planar graph of maximum degree 3 having [Formula: see text] vertices and [Formula: see text] edges has a 2-unit bar visibility representation on [Formula: see text] columns and a 3-connected cubic graph of [Formula: see text] vertices admits a 2-unit bar visibility representation on [Formula: see text] columns.