# Cuts and flows of cell complexes

@article{Duval2012CutsAF, title={Cuts and flows of cell complexes}, author={Art M. Duval and Caroline J. Klivans and Jeremy L. Martin}, journal={Journal of Algebraic Combinatorics}, year={2012}, volume={41}, pages={969-999} }

We study the vector spaces and integer lattices of cuts and flows associated with an arbitrary finite CW complex, and their relationships to group invariants including the critical group of a complex. Our results extend to higher dimension the theory of cuts and flows in graphs, most notably the work of Bacher, de la Harpe, and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and give sufficient conditions for them to be integral…

## 32 Citations

### Cuts and Flows of Cell Complexes Art

- 2013

We study the vector spaces and integer lattices of cuts and flows of an arbitrary finite CW complex, and their relationships to its critical group and related invariants. Our results extend the…

### Simplicial and Cellular Trees

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Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory…

### Simplicial and Cellular Trees Art

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- 2015

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory…

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This work considers high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and gives a combinatorial definition of a simplicial cut that seems more natural in the context of optimization problems and shows that computing such a cut is NP-hard.

### A Simplicial Tutte "5"-flow Conjecture

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This paper concerns a generalization of nowhere-zero modular q-flows from graphs to simplicial complexes of dimension d greater than 1. A modular q-flow of a simplicial complex is an element of the…

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The algorithmic study of some cut problems in high dimensions, namely, Topological Hitting Set and Boundary Nontrivialization, and randomized (approximation) FPT algorithms for the global variants of THS and BNT are initiated.

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In this note we provide a higher-dimensional analogue of Tutte's celebrated theorem on colorings and flows of graphs, by showing that the theory of arithmetic Tutte polynomials and quasi-polynomials…

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We deduce a structurally inductive description of the determinantal probability measure associated with Kalai’s celebrated enumeration result for higher–dimensional spanning trees of the n −…

### Products of arithmetic matroids and quasipolynomial invariants of CW-complexes

- MathematicsJ. Comb. Theory, Ser. A
- 2018

### A family of matrix-tree multijections

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- 2021

. For a natural class of r × n integer matrices, we construct a non-convex polytope which periodically tiles R n . From this tiling, we provide a family of geometrically meaningful maps from a…

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