### Abstract

Let G bean Eulerian digraph, and let {x_{1},x_{2}}, {y_{1}, y_{2}} be two pairs of vertices in G. A directed path from a vertex s to a vertex t is called an st-path. An instance (G; {x_{1}, x_{2}}, {y_{1}, y_{2}}) is called feasible if there is a choice of h,i,j,k with {h,i} = {j,k} = {1,2} such that G has two arc-disjoint X_{h}X_{i}- and y_{j}y_{k}-paths. In this paper, we characterize the structure of minimal infeasible instances, based on which an O(m + n log n) time algorithm is presented to decide whether a given instance is feasible, where n and m are the number of vertices and arcs in the instance, respectively. If the instance is feasible, the corresponding two arc-disjoint paths can be computed in O(m(m + n log n)) time.

Original language | English |
---|---|

Pages (from-to) | 557-589 |

Number of pages | 33 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 11 |

Issue number | 4 |

Publication status | Published - Nov 1998 |

### Fingerprint

### Keywords

- Disjoint paths
- Eulerian digraph
- Minimum cut
- Planar graph
- Polynomial time algorithm

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*SIAM Journal on Discrete Mathematics*,

*11*(4), 557-589.

**Two arc-disjoint paths in Eulerian digraphs.** / Frank, A.; Ibaraki, Toshihide; Nagamochi, Hiroshi.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 11, no. 4, pp. 557-589.

}

TY - JOUR

T1 - Two arc-disjoint paths in Eulerian digraphs

AU - Frank, A.

AU - Ibaraki, Toshihide

AU - Nagamochi, Hiroshi

PY - 1998/11

Y1 - 1998/11

N2 - Let G bean Eulerian digraph, and let {x1,x2}, {y1, y2} be two pairs of vertices in G. A directed path from a vertex s to a vertex t is called an st-path. An instance (G; {x1, x2}, {y1, y2}) is called feasible if there is a choice of h,i,j,k with {h,i} = {j,k} = {1,2} such that G has two arc-disjoint XhXi- and yjyk-paths. In this paper, we characterize the structure of minimal infeasible instances, based on which an O(m + n log n) time algorithm is presented to decide whether a given instance is feasible, where n and m are the number of vertices and arcs in the instance, respectively. If the instance is feasible, the corresponding two arc-disjoint paths can be computed in O(m(m + n log n)) time.

AB - Let G bean Eulerian digraph, and let {x1,x2}, {y1, y2} be two pairs of vertices in G. A directed path from a vertex s to a vertex t is called an st-path. An instance (G; {x1, x2}, {y1, y2}) is called feasible if there is a choice of h,i,j,k with {h,i} = {j,k} = {1,2} such that G has two arc-disjoint XhXi- and yjyk-paths. In this paper, we characterize the structure of minimal infeasible instances, based on which an O(m + n log n) time algorithm is presented to decide whether a given instance is feasible, where n and m are the number of vertices and arcs in the instance, respectively. If the instance is feasible, the corresponding two arc-disjoint paths can be computed in O(m(m + n log n)) time.

KW - Disjoint paths

KW - Eulerian digraph

KW - Minimum cut

KW - Planar graph

KW - Polynomial time algorithm

UR - http://www.scopus.com/inward/record.url?scp=11744320271&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=11744320271&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:11744320271

VL - 11

SP - 557

EP - 589

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 4

ER -