### Abstract

Colin de Verdière introduced an interesting linear algebraic invariant μ(G) of graphs. He proved that μ(G) ≤ 2 if and only if G is outerplanar, and μ(G) ≤ 3 if and only if G is planar. We prove that if the complement of a graph G on n nodes is outerplanar, then μ(G) ≥ n - 4, and if it is planar, then μ(G) ≥ n - 5. We give a full characterization of maximal planar graphs whose complements G have μ(G) = n - 5. In the opposite direction we show that if G does not have "twin" nodes, then μ(G) ≥ n - 3 implies that the complement of G is outerplanar, and μ(G) ≥ n - 4 implies that the complement of G is planar. Our main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks. In particular we show that such sphere representations characterize outerplanar and planar graphs.

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

Pages (from-to) | 483-521 |

Number of pages | 39 |

Journal | Combinatorica |

Volume | 17 |

Issue number | 4 |

Publication status | Published - 1997 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

*17*(4), 483-521.

**The Colin de Verdière number and sphere representations of a graph.** / Kotlov, Andrew; Lovász, L.; Vempala, Santosh.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 17, no. 4, pp. 483-521.

}

TY - JOUR

T1 - The Colin de Verdière number and sphere representations of a graph

AU - Kotlov, Andrew

AU - Lovász, L.

AU - Vempala, Santosh

PY - 1997

Y1 - 1997

N2 - Colin de Verdière introduced an interesting linear algebraic invariant μ(G) of graphs. He proved that μ(G) ≤ 2 if and only if G is outerplanar, and μ(G) ≤ 3 if and only if G is planar. We prove that if the complement of a graph G on n nodes is outerplanar, then μ(G) ≥ n - 4, and if it is planar, then μ(G) ≥ n - 5. We give a full characterization of maximal planar graphs whose complements G have μ(G) = n - 5. In the opposite direction we show that if G does not have "twin" nodes, then μ(G) ≥ n - 3 implies that the complement of G is outerplanar, and μ(G) ≥ n - 4 implies that the complement of G is planar. Our main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks. In particular we show that such sphere representations characterize outerplanar and planar graphs.

AB - Colin de Verdière introduced an interesting linear algebraic invariant μ(G) of graphs. He proved that μ(G) ≤ 2 if and only if G is outerplanar, and μ(G) ≤ 3 if and only if G is planar. We prove that if the complement of a graph G on n nodes is outerplanar, then μ(G) ≥ n - 4, and if it is planar, then μ(G) ≥ n - 5. We give a full characterization of maximal planar graphs whose complements G have μ(G) = n - 5. In the opposite direction we show that if G does not have "twin" nodes, then μ(G) ≥ n - 3 implies that the complement of G is outerplanar, and μ(G) ≥ n - 4 implies that the complement of G is planar. Our main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks. In particular we show that such sphere representations characterize outerplanar and planar graphs.

UR - http://www.scopus.com/inward/record.url?scp=0039464566&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039464566&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0039464566

VL - 17

SP - 483

EP - 521

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 4

ER -