Cones of closed alternating walks and trails
Consider a graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges. The set of all such assignments forms a convex polyhedral co...
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| Published in: | Linear algebra and its applications Vol. 423; no. 2; pp. 351 - 365 |
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01.06.2007
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| ISSN: | 0024-3795 |
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| Abstract | Consider a graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges. The set of all such assignments forms a convex polyhedral cone in the edge space, called the
alternating cone. The integral (respectively, {0,
1}) vectors in the alternating cone are sums of characteristic vectors of closed alternating walks (respectively, trails). We study the basic properties of the alternating cone, determine its dimension and extreme rays, and relate its dimension to the majorization order on degree sequences. We consider whether the alternating cone has integral vectors in a given box, and use residual graph techniques to reduce this problem to the one of searching for an alternating trail connecting two given vertices. The latter problem, called
alternating reachability, is solved in a companion paper along with related results. |
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| AbstractList | Consider a graph whose edges have been colored red and blue. Assign a nonnegative real weight to every edge so that at every vertex, the sum of the weights of the incident red edges equals the sum of the weights of the incident blue edges. The set of all such assignments forms a convex polyhedral cone in the edge space, called the
alternating cone. The integral (respectively, {0,
1}) vectors in the alternating cone are sums of characteristic vectors of closed alternating walks (respectively, trails). We study the basic properties of the alternating cone, determine its dimension and extreme rays, and relate its dimension to the majorization order on degree sequences. We consider whether the alternating cone has integral vectors in a given box, and use residual graph techniques to reduce this problem to the one of searching for an alternating trail connecting two given vertices. The latter problem, called
alternating reachability, is solved in a companion paper along with related results. |
| Author | Peled, Uri N. Srinivasan, Murali K. Bhattacharya, Amitava |
| Author_xml | – sequence: 1 givenname: Amitava surname: Bhattacharya fullname: Bhattacharya, Amitava email: amitava@math.uic.edu organization: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA – sequence: 2 givenname: Uri N. surname: Peled fullname: Peled, Uri N. email: uripeled@uic.edu organization: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA – sequence: 3 givenname: Murali K. surname: Srinivasan fullname: Srinivasan, Murali K. email: mks@math.iitb.ac.in organization: Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India |
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| Keywords | Alternating walks and trails 05C70 90C27 90C57 Colored graphs Cone Vertex(graph) Graph theory 05C70; 90C27; 90C57 Combinatorial optimization Reachability Polyhedron Upper bound Extreme ray Graph colouring Colored graphs; Alternating walks and trails Summation Mathematical programming |
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| References | A. Bhattacharya, U.N. Peled, M.K. Srinivasan, Alternating reachability Arikati, Peled (bib1) 1994; 199 Grossman, Kulkarni, Schochetman (bib4) 1994; 212/213 Marshall, Olkin (bib7) 1979 Ruch, Gutman (bib8) 1979; 4 Ford, Fulkerson (bib3) 1962 Hammer, Ibaraki, Peled (bib5) 1981; vol. 11 Mahadev, Peled (bib6) 1995; vol. 56 Seymour (bib9) 1979 . Ruch (10.1016/j.laa.2007.01.013_bib8) 1979; 4 Mahadev (10.1016/j.laa.2007.01.013_bib6) 1995; vol. 56 Ford (10.1016/j.laa.2007.01.013_bib3) 1962 Arikati (10.1016/j.laa.2007.01.013_bib1) 1994; 199 Hammer (10.1016/j.laa.2007.01.013_bib5) 1981; vol. 11 Seymour (10.1016/j.laa.2007.01.013_bib9) 1979 Marshall (10.1016/j.laa.2007.01.013_bib7) 1979 10.1016/j.laa.2007.01.013_bib2 Grossman (10.1016/j.laa.2007.01.013_bib4) 1994; 212/213 |
| References_xml | – year: 1962 ident: bib3 article-title: Flows in Networks – start-page: 341 year: 1979 end-page: 355 ident: bib9 article-title: Sums of circuits publication-title: Graph Theory and Related Topics – volume: vol. 11 start-page: 125 year: 1981 end-page: 145 ident: bib5 article-title: Threshold numbers and threshold completions publication-title: Studies in Graphs and Discrete Programming – volume: 4 start-page: 286 year: 1979 end-page: 295 ident: bib8 article-title: The branching extent of graphs publication-title: J. Combin. Inform. System Sci. – volume: 199 start-page: 179 year: 1994 end-page: 211 ident: bib1 article-title: Degree sequences and majorization publication-title: Linear Algebra Appl. – reference: A. Bhattacharya, U.N. Peled, M.K. Srinivasan, Alternating reachability, – volume: 212/213 start-page: 289 year: 1994 end-page: 308 ident: bib4 article-title: Algebraic graph theory without orientation publication-title: Linear Algebra Appl. – volume: vol. 56 year: 1995 ident: bib6 article-title: Threshold graphs and related topics publication-title: Annals Discrete Mathematics – reference: . – year: 1979 ident: bib7 article-title: Inequalities: Theory of Majorization and its Applications – volume: vol. 11 start-page: 125 year: 1981 ident: 10.1016/j.laa.2007.01.013_bib5 article-title: Threshold numbers and threshold completions – volume: 199 start-page: 179 year: 1994 ident: 10.1016/j.laa.2007.01.013_bib1 article-title: Degree sequences and majorization publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(94)90349-2 – volume: 212/213 start-page: 289 year: 1994 ident: 10.1016/j.laa.2007.01.013_bib4 article-title: Algebraic graph theory without orientation publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(94)90407-3 – start-page: 341 year: 1979 ident: 10.1016/j.laa.2007.01.013_bib9 article-title: Sums of circuits – volume: vol. 56 year: 1995 ident: 10.1016/j.laa.2007.01.013_bib6 article-title: Threshold graphs and related topics – year: 1979 ident: 10.1016/j.laa.2007.01.013_bib7 – ident: 10.1016/j.laa.2007.01.013_bib2 – year: 1962 ident: 10.1016/j.laa.2007.01.013_bib3 – volume: 4 start-page: 286 year: 1979 ident: 10.1016/j.laa.2007.01.013_bib8 article-title: The branching extent of graphs publication-title: J. Combin. Inform. System Sci. |
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| SubjectTerms | Algebra Alternating walks and trails Applied sciences Colored graphs Combinatorics Combinatorics. Ordered structures Exact sciences and technology Flows in networks. Combinatorial problems Graph theory Linear and multilinear algebra, matrix theory Mathematics Operational research and scientific management Operational research. Management science Sciences and techniques of general use |
| Title | Cones of closed alternating walks and trails |
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