TY - JOUR

T1 - Min-Cost-Flow Preserving Bijection Between Subgraphs and Orientations

AU - Elmaleh, Izhak

AU - Feldheim, Ohad N.

N1 - Publisher Copyright:
© The authors.

PY - 2023

Y1 - 2023

N2 - Consider an undirected graph G = (V, E). A subgraph of G is a subset of its edges, while an orientation of G is an assignment of a direction to each of its edges. Provided with an integer circulation–demand d: V → Z, we show an explicit and efficiently computable bijection between subgraphs of G on which a d-flow exists and orientations on which a d-flow exists. Moreover, given a cost function w: E → (0, ∞) we can find such a bijection which preserves the w-min-cost-flow. In 2013, Kozma and Moran [Electron. J. Comb. 20(3)] showed, using dimensional methods, that the number of subgraphs k-edge-connecting a vertex s to a vertex t is the same as the number of orientations k-edge-connecting s to t. An application of our result is an efficient, bijective proof of this fact.

AB - Consider an undirected graph G = (V, E). A subgraph of G is a subset of its edges, while an orientation of G is an assignment of a direction to each of its edges. Provided with an integer circulation–demand d: V → Z, we show an explicit and efficiently computable bijection between subgraphs of G on which a d-flow exists and orientations on which a d-flow exists. Moreover, given a cost function w: E → (0, ∞) we can find such a bijection which preserves the w-min-cost-flow. In 2013, Kozma and Moran [Electron. J. Comb. 20(3)] showed, using dimensional methods, that the number of subgraphs k-edge-connecting a vertex s to a vertex t is the same as the number of orientations k-edge-connecting s to t. An application of our result is an efficient, bijective proof of this fact.

UR - http://www.scopus.com/inward/record.url?scp=85146157261&partnerID=8YFLogxK

U2 - 10.37236/10940

DO - 10.37236/10940

M3 - Article

AN - SCOPUS:85146157261

SN - 1077-8926

VL - 30

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

IS - 1

M1 - P1.9

ER -