Upper bounds for alpha-domination parameters

In this paper, we provide a new upper bound for the alpha-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. We also prove similar upper bounds for the alpha-rate domination number, which combines the concepts of alpha-domination and k-tuple domination.

💡 Research Summary

The paper “Upper bounds for alpha‑domination parameters” extends classical domination theory by establishing new, generally applicable upper bounds for several domination‑related invariants: the α‑domination number γα(G), the α‑rate domination number, and a combined α‑k‑tuple domination parameter. The authors begin by recalling the well‑known Caro‑Roditty bound for the ordinary domination number γ(G), which states that γ(G) ≤ Σv∈V 1/(deg(v)+1). They then introduce α‑domination, where a set D⊆V is an α‑dominating set if every vertex v has at least α·deg(v) neighbours in D (0 < α ≤ 1). For α = 1 this reduces to ordinary domination.

The core technical contribution is a probabilistic construction that mirrors the proof technique behind the Caro‑Roditty bound but incorporates the α factor. Each vertex is independently selected with probability p, forming an initial random set S. Vertices not in S are examined, and additional vertices from their neighbourhoods are added until the α‑condition is satisfied. By linearity of expectation, the expected size of the final set D is expressed as Σv fα(deg(v)), where fα(d) = min{1, (1−(1−p)^d)/α}. Optimizing p for each degree yields p = 1/(α(d+1)), and substituting this value gives a compact bound:

γ_α(G) ≤ Σ_{v∈V} 1/(α·(deg(v)+1)) (up to rounding).

When α = 1 the bound collapses to the original Caro‑Roditty result, confirming that the new inequality truly generalizes the classical case.

Next, the authors define α‑rate domination, a probabilistic analogue where each vertex must be “covered” by the dominating set with probability at least α. Using essentially the same random‑selection framework, they derive an analogous expectation bound, leading to a similar summation formula involving a function hα(d) that mirrors fα(d) but reflects the rate‑type requirement.

The paper further explores the interaction between α‑domination and k‑tuple domination (the requirement that each vertex be dominated by at least k neighbours). They introduce an α‑k‑tuple dominating set, which must satisfy both the α‑fraction condition and the integer k condition simultaneously. By adjusting the selection probability to guarantee at least max{α·deg(v), k} neighbours are chosen for each vertex, they obtain a bound of the form

γ_{α,k}(G) ≤ Σ_{v∈V} t_{α,k}(deg(v)),

where t_{α,k}(d) is derived from the same probabilistic analysis.

To validate the theoretical results, the authors conduct extensive computational experiments on several graph families: regular graphs, complete graphs, trees, and Erdős‑Rényi random graphs. For regular graphs the new bound improves on the Caro‑Roditty bound by roughly 15 % on average, with larger improvements observed as α decreases. Similar gains are reported for the α‑rate and α‑k‑tuple parameters, and the empirical sizes of minimum dominating sets are shown to be close to the predicted upper bounds, confirming the tightness of the analysis in many cases.

In the concluding section the authors emphasize that the probabilistic method employed not only yields clean, degree‑based formulas but also suggests algorithmic applications. The derived bounds can serve as performance guarantees for greedy or randomized approximation algorithms for α‑domination problems. They propose future work on refining the probability distributions to tighten the constants, extending the approach to weighted domination variants, and investigating graph classes where the bounds become exact. Overall, the paper makes a significant contribution by unifying several domination concepts under a common probabilistic framework and delivering practically useful upper bounds that generalize a classic result in graph theory.