### Abstract

A planar graph G is delta-wye "Δ-Y" reducible if G can be reduced to an edge by a sequence of Δ-Y, series, parallel and degree-1 reductions. Politof characterizes Δ-Y reducible graphs in terms of forbidden homeomorphic subgraphs. A wye-delta "Y-Δ" reducible graph is one that can be reduced to an edge by a sequence of Y-Δ, series, parallel and degree-1 reductions. Y-Δ reducible graphs are all partial 3-trees. Recently, Arnborg and Proskurowski have shown confluent reductions which are both necessary and sufficient for the recognition of partial 3-trees. In this paper we note that Δ-Y graphs are the planar duals of Y-Δ graphs. We exploit this duality and the known reduction rules for partial 3-trees to characterize both classes of graphs using forbidden minors. The result yields a shorter proof of Politof's result. In addition, we give linear time algorithms for recognizing such graphs and for embedding any Δ-Y graph in a 4-tree. These algorithms complement many known linear time algorithms for solving some hard network problems on graphs given their embedding in a k-tree for some fixed k.

Original language | English (US) |
---|---|

Pages (from-to) | 21-40 |

Number of pages | 20 |

Journal | Discrete Mathematics |

Volume | 80 |

Issue number | 1 |

DOIs | |

State | Published - Feb 15 1990 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*80*(1), 21-40. https://doi.org/10.1016/0012-365X(90)90293-Q

**On two dual classes of planar graphs.** / El-Mallah, Ehab S.; Colbourn, Charles.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 80, no. 1, pp. 21-40. https://doi.org/10.1016/0012-365X(90)90293-Q

}

TY - JOUR

T1 - On two dual classes of planar graphs

AU - El-Mallah, Ehab S.

AU - Colbourn, Charles

PY - 1990/2/15

Y1 - 1990/2/15

N2 - A planar graph G is delta-wye "Δ-Y" reducible if G can be reduced to an edge by a sequence of Δ-Y, series, parallel and degree-1 reductions. Politof characterizes Δ-Y reducible graphs in terms of forbidden homeomorphic subgraphs. A wye-delta "Y-Δ" reducible graph is one that can be reduced to an edge by a sequence of Y-Δ, series, parallel and degree-1 reductions. Y-Δ reducible graphs are all partial 3-trees. Recently, Arnborg and Proskurowski have shown confluent reductions which are both necessary and sufficient for the recognition of partial 3-trees. In this paper we note that Δ-Y graphs are the planar duals of Y-Δ graphs. We exploit this duality and the known reduction rules for partial 3-trees to characterize both classes of graphs using forbidden minors. The result yields a shorter proof of Politof's result. In addition, we give linear time algorithms for recognizing such graphs and for embedding any Δ-Y graph in a 4-tree. These algorithms complement many known linear time algorithms for solving some hard network problems on graphs given their embedding in a k-tree for some fixed k.

AB - A planar graph G is delta-wye "Δ-Y" reducible if G can be reduced to an edge by a sequence of Δ-Y, series, parallel and degree-1 reductions. Politof characterizes Δ-Y reducible graphs in terms of forbidden homeomorphic subgraphs. A wye-delta "Y-Δ" reducible graph is one that can be reduced to an edge by a sequence of Y-Δ, series, parallel and degree-1 reductions. Y-Δ reducible graphs are all partial 3-trees. Recently, Arnborg and Proskurowski have shown confluent reductions which are both necessary and sufficient for the recognition of partial 3-trees. In this paper we note that Δ-Y graphs are the planar duals of Y-Δ graphs. We exploit this duality and the known reduction rules for partial 3-trees to characterize both classes of graphs using forbidden minors. The result yields a shorter proof of Politof's result. In addition, we give linear time algorithms for recognizing such graphs and for embedding any Δ-Y graph in a 4-tree. These algorithms complement many known linear time algorithms for solving some hard network problems on graphs given their embedding in a k-tree for some fixed k.

UR - http://www.scopus.com/inward/record.url?scp=38249020050&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249020050&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(90)90293-Q

DO - 10.1016/0012-365X(90)90293-Q

M3 - Article

AN - SCOPUS:38249020050

VL - 80

SP - 21

EP - 40

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1

ER -