Monotone Drawings of $k$-Inner Planar Graphs

TL;DR

This paper presents a method to construct monotone drawings of $k$-inner planar graphs on polynomial-sized grids, with improved results for outerplanar graphs, enhancing visualization techniques for these graph classes.

Contribution

The paper introduces a new construction method for monotone drawings of $k$-inner planar graphs on polynomial grids, improving existing results for outerplanar graphs.

Findings

Monotone drawings of $k$-inner planar graphs can be achieved on a $2(k+1)n imes 2(k+1)n$ grid.

Outerplanar graphs admit planar monotone drawings on an $n imes n$ grid, improving previous bounds.

The method enhances graph visualization by providing efficient grid drawings for these classes.

Abstract

A $k$-inner planar graph is a planar graph that has a plane drawing with at most $k$ {internal vertices}, i.e., vertices that do not lie on the boundary of the outer face of its drawing. An outerplanar graph is a $0$-inner planar graph. In this paper, we show how to construct a monotone drawing of a $k$-inner planar graph on a $2(k+1)n \times 2(k+1)n$ grid. In the special case of an outerplanar graph, we can produce a planar monotone drawing on a $n \times n$ grid, improving previously known results.