An Orientation Theorem with Parity Conditions
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
An Orientation Theorem with Parity Conditions
Proceedings of the 7th International IPCO Conference on Integer Programming and Combinatorial Optimization
Combined Connectivity Augmentation and Orientation Problems
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
The edge-orientation problem and some of its variants on weighted graphs
Information Sciences: an International Journal
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Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V,E) having a k-edge-connected T-odd orientation for every subset T ⊆ V with |E|+|T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k + 2)-edge-connected graph with |V|+|E| even has a k-edge-connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k+1 edge-disjoint paths from s to every node and k edge-disjoint paths from every node to s.