### Abstract

A simple undirected graph G=(V,E) is a rigidity circuit if |E|=2|V|-2 and |E _{G}[X]|≤2|X|-3 for every X⊂V with 2≤|X|≤|V|-1, where E _{G}[X] denotes the set of edges connecting vertices in X. It is known that a rigidity circuit can be decomposed into two edge-disjoint spanning trees. Graver et al. (1993) [5] asked if any rigidity circuit with maximum degree 4 can be decomposed into two edge-disjoint Hamiltonian paths. This paper presents infinitely many counterexamples for the question. Counterexamples are constructed based on a new characterization of a 3-connected plane graph in terms of the sparsity of its medial graph and a sufficient condition for the connectivity of medial graphs.

Original language | English |
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Pages (from-to) | 2466-2472 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 312 |

Issue number | 16 |

DOIs | |

Publication status | Published - Aug 28 2012 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*312*(16), 2466-2472. https://doi.org/10.1016/j.disc.2012.04.013

**Sparsity and connectivity of medial graphs : Concerning two edge-disjoint Hamiltonian paths in planar rigidity circuits.** / Kijima, Shuji; Tanigawa, Shin Ichi.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 312, no. 16, pp. 2466-2472. https://doi.org/10.1016/j.disc.2012.04.013

}

TY - JOUR

T1 - Sparsity and connectivity of medial graphs

T2 - Concerning two edge-disjoint Hamiltonian paths in planar rigidity circuits

AU - Kijima, Shuji

AU - Tanigawa, Shin Ichi

PY - 2012/8/28

Y1 - 2012/8/28

N2 - A simple undirected graph G=(V,E) is a rigidity circuit if |E|=2|V|-2 and |E G[X]|≤2|X|-3 for every X⊂V with 2≤|X|≤|V|-1, where E G[X] denotes the set of edges connecting vertices in X. It is known that a rigidity circuit can be decomposed into two edge-disjoint spanning trees. Graver et al. (1993) [5] asked if any rigidity circuit with maximum degree 4 can be decomposed into two edge-disjoint Hamiltonian paths. This paper presents infinitely many counterexamples for the question. Counterexamples are constructed based on a new characterization of a 3-connected plane graph in terms of the sparsity of its medial graph and a sufficient condition for the connectivity of medial graphs.

AB - A simple undirected graph G=(V,E) is a rigidity circuit if |E|=2|V|-2 and |E G[X]|≤2|X|-3 for every X⊂V with 2≤|X|≤|V|-1, where E G[X] denotes the set of edges connecting vertices in X. It is known that a rigidity circuit can be decomposed into two edge-disjoint spanning trees. Graver et al. (1993) [5] asked if any rigidity circuit with maximum degree 4 can be decomposed into two edge-disjoint Hamiltonian paths. This paper presents infinitely many counterexamples for the question. Counterexamples are constructed based on a new characterization of a 3-connected plane graph in terms of the sparsity of its medial graph and a sufficient condition for the connectivity of medial graphs.

UR - http://www.scopus.com/inward/record.url?scp=84861867158&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861867158&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2012.04.013

DO - 10.1016/j.disc.2012.04.013

M3 - Article

AN - SCOPUS:84861867158

VL - 312

SP - 2466

EP - 2472

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 16

ER -