ISCO 2012: Combinatorial Optimization pp 356-367

# The Gram Dimension of a Graph

• Monique Laurent
• Antonios Varvitsiotis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)

## Abstract

The Gram dimension $$\text{\rm gd}(G)$$ 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 ℝ k , having the same inner products on the edges of the graph. The class of graphs satisfying $$\text{\rm gd}(G) \le k$$ 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 k + 1. We show that a graph has Gram dimension at most 4 if and only if it does not have K 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].

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## Authors and Affiliations

• Monique Laurent
• 1
• 2
• Antonios Varvitsiotis
• 1
1. 1.Centrum Wiskunde & Informatica (CWI)AmsterdamThe Netherlands
2. 2.Tilburg UniversityThe Netherlands