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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Bibliographic Details
Published in:Linear algebra and its applications Vol. 423; no. 2; pp. 351 - 365
Main Authors: Bhattacharya, Amitava, Peled, Uri N., Srinivasan, Murali K.
Format: Journal Article
Language:English
Published: New York, NY Elsevier Inc 01.06.2007
Elsevier Science
Subjects:
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
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
ISSN:0024-3795
Online Access:Get full text
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Summary: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.
ISSN:0024-3795
DOI:10.1016/j.laa.2007.01.013