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VPG and EPG bend-numbers of Halin graphs

Francis, Mathew C and Lahiri, Abhiruk (2016) VPG and EPG bend-numbers of Halin graphs. In: DISCRETE APPLIED MATHEMATICS, 215 . pp. 95-105. PDF Dis_App_Mat_215_95_2016.pdf - Published Version Restricted to Registered users only Download (512kB) | Request a copy
Official URL: http://dx.doi.org/10.1016/j.dam.2016.07.007

Abstract

A piecewise linear simple 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 Bk-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) <= 1 and b(e)(G) <= 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. (C) 2016 Elsevier B.V. All rights reserved.

Item Type: Journal Article Copy right for this article belongs to the ELSEVIER SCIENCE BV, PO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS Division of Electrical Sciences > Computer Science & Automation Id for Latest eprints 03 Dec 2016 06:33 03 Dec 2016 06:33 http://eprints.iisc.ac.in/id/eprint/55298 View Item