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Recognizing Distance-Count Matrices is Difficult

Paolo Boldi, Flavio Furia, Chiara Prezioso, Ian Stewart

TL;DR

This work studies the distance-count matrix (DCM) as a compact descriptor of vertex distance distributions that underpin geometric centralities. It proves that recognizing whether a given matrix is a DCM (and its cumulative variant CDCM) is strongly NP-complete via a reduction from a specially constrained three-partition problem, constructing a matrix $M(\mathbf{a})$ from a TPP instance and a corresponding graph $\mathcal{G}(\mathbf{a})$ to establish a tight equivalence. The result implies that brute-force search for DCM-based counterexamples to centrality properties is impractical, guiding researchers toward alternative strategies or identifying tractable graph families. The paper also clarifies the relationship between DCMs, distance degree sequences, and graphical realization concepts, and outlines directions for future work on efficiently recognizable subfamilies and preservation under graph operations.

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.

Recognizing Distance-Count Matrices is Difficult

TL;DR

This work studies the distance-count matrix (DCM) as a compact descriptor of vertex distance distributions that underpin geometric centralities. It proves that recognizing whether a given matrix is a DCM (and its cumulative variant CDCM) is strongly NP-complete via a reduction from a specially constrained three-partition problem, constructing a matrix from a TPP instance and a corresponding graph to establish a tight equivalence. The result implies that brute-force search for DCM-based counterexamples to centrality properties is impractical, guiding researchers toward alternative strategies or identifying tractable graph families. The paper also clarifies the relationship between DCMs, distance degree sequences, and graphical realization concepts, and outlines directions for future work on efficiently recognizable subfamilies and preservation under graph operations.

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.

Paper Structure

This paper contains 9 sections, 9 theorems, 10 equations, 3 figures.

Table of Contents

  1. Introduction and Related Work
  2. Notation
  3. Distance-count matrices (DCM)
  4. Graphical sequences for undirected graphs
  5. Graphical in-degree sequences
  6. Some basic properties of (C)DCMs
  7. NP-Completeness of (C)DCM recognizability
  8. The three-partition problem
  9. The three-partition problem

Key Result

theorem thmcountertheorem

Consider a sequence $d_1 \geq d_2 \geq \dots \geq d_p$ of $p$ positive integers. This sequence is a graphical degree sequence if and only if its sum is even and for every $n=1,\dots,p-1$

Figures (3)

  • Figure 1: A strongly connected graph ${\mathcal{G}}$.
  • Figure 2: The DCM $N_{\mathcal{G}}$ and CDCM $M_{\mathcal{G}}$ for the graph ${\mathcal{G}}$ of Figure \ref{['fig:graph']}. The highlighted rows correspond to $n_{{\mathcal{G}},-}(1)$ and $n_{{\mathcal{G}},-}(3)$.
  • Figure 3: Construction for a graph with a given good row.

Theorems & Definitions (16)

  • definition thmcounterdefinition: DCM and CDCM
  • theorem thmcountertheorem: erdHos1960grafok
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proof
  • corollary thmcountercorollary
  • definition thmcounterdefinition: Good sequence
  • ...and 6 more