Alpha-Beta Duality for Continuous Graphs

Prove or refute the conjecture that every continuous graph Γ with n endpoints satisfies α₁(Γ)+β₁(Γ)=n, and determine a tight bound on αᵣ(Γ)+βᵣ(Γ) in terms of n and r for arbitrary r>0.

Background

In classical graph theory, the independence number and vertex-cover number sum to the number of vertices. The paper investigates whether an analogous identity holds for continuous graphs when both points of independent sets and centers of covering balls may lie anywhere on the graph. Experiments and an example support the r=1 identity, but no proof is provided; the authors also ask for the general-radius bound and its tightness.

References

Thus, we conjecture that for any continuous graph $\Gamma$, $\alpha_1(\Gamma)+\beta_1(\Gamma)=n$. In general, the following question can be addressed: Given $r>0$, what is the bound on $\alpha_r(\Gamma)+\beta_r(\Gamma)$ for a continuous graph $\Gamma$ in terms of $n$ and $r$? Is the bound tight?

— Open Problems in Continuous Graphs  (2501.14554 - Grigoriev et al., 24 Jan 2025) in Section 3, subsection “Vertex cover,” paragraph “The α-β duality”