### Abstract

*G)*of a graph is the smallest integer

*k*≥ 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in ℝ

^{k}, having the same inner products on the edges of the graph. The class of graphs satisfying \rm gd

*(G)*≤

*k*is minor closed for fixed

*k*, so it can characterized by a finite list of forbidden minors. For

*k*≤ 3, the only forbidden minor is

*K*. We show that a graph has Gram dimension at most 4 if and only if it does not have K

_{k + 1}_{5}and K

_{2,2,2}as minors. We also show some close connections to the notion of

*d*-realizability of graphs. In particular, our result implies the characterization of 3-realizable graphs of Belk and Connelly [5,6].

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

Title of host publication | Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO 2012) |

Editors | A.R. Mahjoub, V. Markakis, I. Milis, V. Paschos |

Place of Publication | Berlin Heidelberg |

Publisher | Springer Verlag |

Pages | 356-367 |

Volume | 7422 |

ISBN (Print) | 9783642321467 |

Publication status | Published - 2012 |

### Publication series

Name | Lecture Notes in Computer Science |
---|---|

Volume | 7422 |

### Fingerprint

### Cite this

*Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO 2012)*(Vol. 7422, pp. 356-367). (Lecture Notes in Computer Science; Vol. 7422). Berlin Heidelberg: Springer Verlag.

}

*Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO 2012).*vol. 7422, Lecture Notes in Computer Science, vol. 7422, Springer Verlag, Berlin Heidelberg, pp. 356-367.

**The gram dimension of a graph.** / Laurent, M.; Varvitsiotis, A.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Scientific › peer-review

TY - GEN

T1 - The gram dimension of a graph

AU - Laurent, M.

AU - Varvitsiotis, A.

PY - 2012

Y1 - 2012

N2 - The Gram dimension \rm gd(G) of a graph is the smallest integer k ≥ 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in ℝk, having the same inner products on the edges of the graph. The class of graphs satisfying \rm gd(G) ≤ k is minor closed for fixed k, so it can characterized by a finite list of forbidden minors. For k ≤ 3, the only forbidden minor is Kk + 1. We show that a graph has Gram dimension at most 4 if and only if it does not have K5 and K2,2,2 as minors. We also show some close connections to the notion of d-realizability of graphs. In particular, our result implies the characterization of 3-realizable graphs of Belk and Connelly [5,6].

AB - The Gram dimension \rm gd(G) of a graph is the smallest integer k ≥ 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in ℝk, having the same inner products on the edges of the graph. The class of graphs satisfying \rm gd(G) ≤ k is minor closed for fixed k, so it can characterized by a finite list of forbidden minors. For k ≤ 3, the only forbidden minor is Kk + 1. We show that a graph has Gram dimension at most 4 if and only if it does not have K5 and K2,2,2 as minors. We also show some close connections to the notion of d-realizability of graphs. In particular, our result implies the characterization of 3-realizable graphs of Belk and Connelly [5,6].

M3 - Conference contribution

SN - 9783642321467

VL - 7422

T3 - Lecture Notes in Computer Science

SP - 356

EP - 367

BT - Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO 2012)

A2 - Mahjoub, A.R.

A2 - Markakis, V.

A2 - Milis, I.

A2 - Paschos, V.

PB - Springer Verlag

CY - Berlin Heidelberg

ER -