Forbidden minors characterization of partial 3-trees
Discrete Mathematics
Graph Minors. XX. Wagner's conjecture
Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
A semidefinite programming approach to tensegrity theory and realizability of graphs
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Discrete & Computational Geometry
Realizability of Graphs in Three Dimensions
Discrete & Computational Geometry
Exact Matrix Completion via Convex Optimization
Foundations of Computational Mathematics
Geometry of Cuts and Metrics
The rotational dimension of a graph
Journal of Graph Theory
Rounding two and three dimensional solutions of the SDP relaxation of MAX CUT
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
On the Shannon capacity of a graph
IEEE Transactions on Information Theory
The max-cut problem on graphs not contractible to K5
Operations Research Letters
| Hi-index | 0.00 |
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 ℝk, 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 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].