Paired domination in trees: A linear algorithm and asymptotic normality

TL;DR

The paper tackles the minimum paired dominating set problem on trees by introducing a linear-time bottom-up algorithm that constructs a $\gamma_{\rm pr}$-set and proves its optimality. It then proves a nondegenerate Gaussian limit law for $\gamma_{\rm pr}$ in conditioned Galton–Watson trees, via an additive-parameter framework and a bounded toll function, with model-specific means $\mu_{\rm pr}$ and positive variance $\sigma_{\rm pr}^2$. The results yield explicit asymptotic means for several tree families (Binary, Plane, and Labeled) and apply to random Cayley trees as well, supported by simulations. Together, the contributions advance both the algorithmic tractability and probabilistic understanding of paired domination in random tree models, with potential applications in graph theory and network design.

Abstract

A set $S$ of vertices in a graph $G$ is a paired dominating set if every vertex of $G$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ contains a perfect matching (not necessarily as an induced subgraph). The paired domination number, $γ_{\mathrm{pr}}(G)$, of $G$ is the minimum cardinality of a paired dominating set of $G$. We present a linear algorithm for computing the paired domination number of a tree. As an application of our algorithm, we prove that the paired domination number is asymptotically normal in a random rooted tree of order $n$ generated by a conditioned Galton-Watson process as $n\to\infty$. In particular, we have found that the paired domination number of a random Cayley tree of order $n$, where each tree is equally likely, is asymptotically normal with expectation approaching $(0.5177\ldots)n$.

Figures (5)

• Figure 1: A rooted tree $T$ with its Parent array
• Figure 2: A tree with labelled vertices
• Figure 3: Fixed tree $\tau_0$ of height $3$.
• Figure 4: Two trees, $t_1$ and $t_2$, of the same size with different paired domination numbers.
• Figure 5: Histogram from $20000$ random samples of Cayley trees of order $n=10000$

Theorems & Definitions (13)

• Theorem 1
• proof
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3: Naina
• Lemma 3
• proof