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arXiv · 2309.04892

Descriptive complexity of controllable graphs

Aida Abiad·Anuj Dawar·Octavio Zapata
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Abstract

Let $G$ be a graph on $n$ vertices with adjacency matrix $A$, and let $\mathbf{1}$ be the all-ones vector. We call $G$ controllable if the set of vectors $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ spans the whole space $\mathbb{R}^n$. We characterize the isomorphism problem of controllable graphs in terms of other combinatorial, geometric and logical problems. We also describe a polynomial time algorithm for graph isomorphism that works for almost all graphs.

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Aida Abiad, Anuj Dawar, Octavio Zapata. 2023-09-09. Descriptive complexity of controllable graphs. https://arxiv.org/abs/2309.04892

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