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arXiv · 2508.18857

Recognizing Distance-Count Matrices is Difficult

Paolo Boldi·Flavio Furia·Chiara Prezioso·Ian Stewart
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Abstract

Axiomatization of centrality measures often involves proving that something cannot hold by providing a counterexample (i.e., a graph for which that specific centrality index fails to have a given property). In the context of geometric centralities, building such counterexamples requires constructing a graph with specific distance counts between nodes, as expressed by its distance-count matrix. We prove that deciding whether a matrix is the distance-count matrix of a graph is strongly NP-complete. This negative result implies that a brute-force approach to building this kind of counterexample is out of question, and cleverer approaches are required.

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Paolo Boldi, Flavio Furia, Chiara Prezioso, Ian Stewart. 2025-08-26. Recognizing Distance-Count Matrices is Difficult. https://arxiv.org/abs/2508.18857

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