# On Integrality, Stability and Composition of Dicycle Packings and Covers

@article{Nutov2000OnIS, title={On Integrality, Stability and Composition of Dicycle Packings and Covers}, author={Zeev Nutov and Michal Penn}, journal={Journal of Combinatorial Optimization}, year={2000}, volume={4}, pages={235-251} }

Given a digraph D, the minimum integral dicycle cover problem (known also as the minimum feedback arc set problem) is to find a minimum set of arcs that intersects every dicycle; the maximum integral dicycle packing problem is to find a maximum set of pairwise arc disjoint dicycles. These two problems are NP-complete.Assume D has a 2-vertex cut. We show how to derive a minimum dicycle cover (a maximum dicycle packing) for D, by composing certain covers (packings) of the corresponding pieces…

## 2 Citations

### Invited review: Utilizing peripheral nerve regenerative elements to repair damage in the CNS

- Biology, MedicineJournal of Neuroscience Methods
- 2020

## References

SHOWING 1-10 OF 33 REFERENCES

### on the Integral Dicycle Packings and Covers and the Linear ordering Polytope

- MathematicsDiscret. Appl. Math.
- 1995

### Compositions of Graphs and Polyhedra IV: Acyclic Spanning Subgraphs

- MathematicsSIAM J. Discret. Math.
- 1994

It is shown that, for graphs with no $K_{3,3}$ minor, the cycle inequalities characterize the acyclic subgraph polytope and form a TDI system and the cardinality of a minimum feedback set is equal to the maximum number of arc disjoint cycles.

### A representation for crossing set families with applications to submodular flow problems

- MathematicsSODA '93
- 1993

A representation for intersecting and crossing families of sets is presented, based on a separator theorem for intersectioning families, which is used to improve several submodular flow algorithms.

### Finding a Minimum Feedback Arc Set in Reducible Flow Graphs

- Computer Science, MathematicsJ. Algorithms
- 1988

### Compositions of Graphs and Polyhedra II: Stable Sets

- MathematicsSIAM J. Discret. Math.
- 1994

A compact system for the stable set problem in series-parallel graphs is derived and this technique is also applied to characterize facet-defining inequalities for graphs with no $K_{5}\e$ minor.

### How to make a digraph strongly connected

- Mathematics, Computer ScienceComb.
- 1981

AnO(n5) primal-dual algorithm is presented for finding a minimum weight covering of an edge-weighted digraph and a constructive proof for a min-max theorem due to Lucchesi and Younger is provided.

### Compositions of Graphs and Polyhedra III: Graphs with No W4 Minor

- MathematicsSIAM J. Discret. Math.
- 1994

By adding some extra variables, it is shown that the stable set problem for these graphs can be formulated as a linear program of polynomial size.

### A Minimax Theorem for Directed Graphs

- Mathematics
- 1978

This minimax equality was conjectured about a decade ago by one of the authors ([7; page 43], [8], [9]) and, independently, by Neil Robertson. It arose in the study of a problem posed several years…

### An extended planar algorithm for maximum integral two-flow

- MathematicsNetworks
- 1998

This paper extends the algorithm for maximum integral two-flow in planar graphs to certain undirected K3,3-free graphs (graphs not containing any subgraph homeomorphic to K 3,3 ) and provides an O (n) algorithm for determining a maximum integralTwo-flow of a value not less than the value of a maximum two- flow minus two.

### Packing directed circuits fractionally

- MathematicsComb.
- 1995

There is a set ofO (k logk log k logk) vertices meeting all directed circuits ofG, such that no “fractional” packing of directed circuit ofG has value >k, when every vertex is given “capacity” 1.