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Structure of betweenness uniform graphs with low values of betweenness centrality

Babak Ghanbari, David Hartman, Vít Jelínek, Aneta Pokorná, Robert Šámal, Pavel Valtr

Abstract

This work deals with undirected graphs that have the same betweenness centrality for each vertex, so-called betweenness uniform graphs (or BUGs). The class of these graphs is not trivial and its classification is still an open problem. Recently, Gago, Coroničová-Hurajová and Madaras conjectured that for every rational $α\ge 3/4$ there exists a BUG having betweenness centrality~$α$. We disprove this conjecture, and provide an alternative view of the structure of betweenness-uniform graphs from the point of view of their complement. This allows us to characterise all the BUGs with betweennes centrality at most 9/10, and show that their betweenness centrality is equal to $\frac{\ell}{\ell+1}$ for some integer $\ell\le 9$. We conjecture that this characterization extends to all the BUGs with betweenness centrality smaller than~1.

Structure of betweenness uniform graphs with low values of betweenness centrality

Abstract

This work deals with undirected graphs that have the same betweenness centrality for each vertex, so-called betweenness uniform graphs (or BUGs). The class of these graphs is not trivial and its classification is still an open problem. Recently, Gago, Coroničová-Hurajová and Madaras conjectured that for every rational there exists a BUG having betweenness centrality~. We disprove this conjecture, and provide an alternative view of the structure of betweenness-uniform graphs from the point of view of their complement. This allows us to characterise all the BUGs with betweennes centrality at most 9/10, and show that their betweenness centrality is equal to for some integer . We conjecture that this characterization extends to all the BUGs with betweenness centrality smaller than~1.
Paper Structure (11 sections, 27 theorems, 29 equations)

This paper contains 11 sections, 27 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and notation
  4. Bounds for betweenness via edge-vertex ratio
  5. Low-betweenness BUGs with diameter at least three
  6. BUGs with diameter two
  7. Low-betweenness BUGs
  8. No exotic coBUGs with uniform components
  9. No exotic BUGs with betweenness up to 9/10
  10. Ruling out small stars in exotic coBUGs
  11. Conclusion and open questions

Key Result

Theorem 3

Let $G$ be a BUG with betweenness $\alpha\le 9/10$. Then $\alpha=\ell/(\ell+1)$ for some integer $\ell\ge 0$, and the complement of $G$ is a graph whose every connected component is isomorphic to $K_{1,\ell}$.

Theorems & Definitions (56)

  • Conjecture 2
  • Theorem 3
  • Lemma 4
  • proof
  • Corollary 5
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • Example 1
  • ...and 46 more