# A note on packing paths in planar graphs

@article{Frank1995ANO,
title={A note on packing paths in planar graphs},
author={Andr{\'a}s Frank and Zolt{\'a}n Szigeti},
journal={Mathematical Programming},
year={1995},
volume={70},
pages={201-209}
}
• Published 30 October 1995
• Mathematics
• Mathematical Programming
Seymour (1981) proved that the cut criterion is necessary and sufficient for the solvability of the edge-disjoint paths problem when the union of the supply graph and the demand graph is planar and Eulerian. When only planarity is required, Middendorf and Pfeiffer (1993) proved the problem to be NP-complete. For this case, Korach and Penn (1992) proved that the cut criterion is sufficient for the existence of a near-complete packing of paths. Here we generalize this result by showing how a…

### The planar tree packing theorem

• Mathematics
J. Comput. Geom.
• 2016
This paper settles the conjecture in the affirmative and proves its general form, thus making it the planar tree packing theorem, and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.

### Potentials in Undirected Graphs and Planar Multiflows

The goal of the present paper is to extrapolate from this $\pm 1$-weighted bipartite special case the arbitrarily weighted general min-path-max-potential theorem and to show some algorithmic consequences related to planar multiflows, the Chinese postman problem, the weighted and unweighted matching structure, etc.

### POTENTIALS IN UNDIRECTED GRAPHS AND PLANAR MULTIFLOWS

The goal of the present paper is to extrapolate from this ±1weighted bipartite special case the arbitrarily weighted general min-path-max-potential theorem and to show some algorithmic consequences related to planar multiflows, the Chinese postman problem, the weighted and unweighted matching structure, etc.

### A Survey On T-joins, T-cuts, and conservative weightings

The book of L. Lovv asz and M. Plummer on matching theory includes a good overview of the topic indicated in the title. In the present paper we exhibit the main developments of the area in the last

### Integer Plane Multiflow Maximisation: Flow-Cut Gap and One-Quarter-Approximation

• Mathematics
IPCO
• 2020
The integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-cut-gap are bound and planarity means here that the union of the supply and demand graph is planar.

### Integer Programming and Combinatorial Optimization: 21st International Conference, IPCO 2020, London, UK, June 8–10, 2020, Proceedings

• Mathematics
IPCO
• 2020
It is conjecture that every 4-wise intersecting clutter is non-ideal, and the proof is proved in the binary case using Jaeger's 8-flow theorem for graphs and Seymour's characterization of the binary matroids with the sums of circuits property.

### Integer plane multiflow maximisation: one-quarter-approximation and gaps

• Mathematics
Mathematical Programming
• 2021
In this paper, we bound the integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-cut-gap. We consider instances where the union of the supply

### Planar Packing of Diameter-Four Trees

We prove that, for every two n-node non-star trees of diameter at most four, there exists an n-node planar graph containing them as edge-disjoint subgraphs.

### Planar Packing of Binary Trees

• Mathematics
• 2013
The proof is algorithmic and yields a linear time algorithm to compute a plane packing, that is, a suitable two-edge-colored host graph along with a planar embedding for it, which can also guarantee several nice geometric properties for the embedding.

## References

SHOWING 1-8 OF 8 REFERENCES

### On the complexity of the disjoint paths problem

• Mathematics
Comb.
• 1993
It is shown that (assumingP≠NP) one can drop neither planarity nor the Eulerian condition onG without losing polynomial time solvability, which implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerians graphs.

### Potentials in Undirected Graphs and Planar Multiflows

The goal of the present paper is to extrapolate from this $\pm 1$-weighted bipartite special case the arbitrarily weighted general min-path-max-potential theorem and to show some algorithmic consequences related to planar multiflows, the Chinese postman problem, the weighted and unweighted matching structure, etc.

### Tight integral duality gap in the Chinese Postman problem

• Mathematics
Math. Program.
• 1992
For a certain integral multicommodity flow problem in the plane, which was recently proved to be NP-complete, the above result gives a solution such that for every commodity the flow is less than the demand by at most one unit.

### On Odd Cuts and Plane Multicommodity Flows

Let T be an even subset of the vertices of a graph G. A T-cut is an edge-cutset of the graph which divides T into two odd sets. We prove that if G is bipartite, then the maximum number of disjoint

### Packing paths, circuits, and cuts -a survey, in: "Paths, Flows and VLSI-Layouts

• Packing paths, circuits, and cuts -a survey, in: "Paths, Flows and VLSI-Layouts

### Graphic programming using odd and even points

• Chinese Journal of Mathematics
• 1962