VPG and EPG bend-numbers of Halin Graphs

TL;DR

This paper establishes tight bounds on the bend-numbers for VPG and EPG representations of Halin graphs, showing they can be represented with at most 1 and 2 bends respectively, and extends these results to certain planar graphs.

Contribution

It proves that Halin graphs have VPG bend-number at most 1 and EPG bend-number at most 2, providing tight bounds and extending to planar graphs formed by connecting leaves of a tree.

Findings

Halin graphs have VPG bend-number ≤ 1.

Halin graphs have EPG bend-number ≤ 2.

These bounds are tight for the classes considered.

Abstract

A piecewise linear curve in the plane made up of $k+1$ line segments, each of which is either horizontal or vertical, with consecutive segments being of different orientation is called a $k$-bend path. Given a graph $G$, a collection of $k$-bend paths in which each path corresponds to a vertex in $G$ and two paths have a common point if and only if the vertices corresponding to them are adjacent in $G$ is called a $B_k$-VPG representation of $G$. Similarly, a collection of $k$-bend paths each of which corresponds to a vertex in $G$ is called an $B_k$-EPG representation of $G$ if any two paths have a line segment of non-zero length in common if and only if their corresponding vertices are adjacent in $G$. The VPG bend-number $b_v(G)$ of a graph $G$ is the minimum $k$ such that $G$ has a $B_k$-VPG representation. Similarly, the EPG bend-number $b_e(G)$ of a graph $G$ is the minimum $k$ such that $G$ has a $B_k$-EPG representation. Halin graphs are the graphs formed by taking a tree with no degree $2$ vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if $G$ is a Halin graph then $b_v(G) \leq 1$ and $b_e(G) \leq 2$. These bounds are tight. In fact, we prove the stronger result that if $G$ is a planar graph formed by connecting the leaves of any tree to form a simple cycle, then it has a VPG-representation using only one type of 1-bend paths and an EPG-representation using only one type of 2-bend paths.