Layouts of graph subdivisions

Vida Dujmović, David R. Wood

Research output: Chapter in Book/Report/Conference proceedingConference PaperResearchpeer-review

3 Citations (Scopus)

Abstract

A k-stack layout (respectively, k-queue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The stack-number (respectively, queue-number, track-number) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout. This paper studies stack, queue, and track layouts of graph sub-divisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from OScript(log n) to OScript(log min{sn(G),qn(G)}). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number. It is proved that every graph has a 2-queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision. All these values are optimal for every non-planar graph. In addition, we characterise those graphs with k-stack, k-queue, and k-track subdivisions, for all values of k. The number of division vertices per edge in the case of 2-queue and 4-track subdivisions, namely OScript(log qn(G)), is optimal to within a constant factor, for every graph G. Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with OScript(log qn(G)) bends per edge.

Original languageEnglish
Title of host publicationGraph Drawing
Subtitle of host publication12th International Symposium, GD 2004 New York, NY, USA, September 29-October 2, 2004 Revised Selected Papers
EditorsJános Pach
Place of PublicationBerlin Germany
PublisherSpringer
Pages133-143
Number of pages11
ISBN (Print)3540245286
DOIs
Publication statusPublished - 1 Dec 2004
Externally publishedYes
EventGraph Drawing 2004 - New York, United States of America
Duration: 29 Sep 20042 Oct 2004
Conference number: 12th
https://link.springer.com/book/10.1007/b105810 (Proceedings)

Publication series

NameLecture Notes in Computer Science
PublisherSpringer
Volume3383
ISSN (Print)0302-9743

Conference

ConferenceGraph Drawing 2004
Abbreviated titleGD 2004
CountryUnited States of America
CityNew York
Period29/09/042/10/04
Internet address

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