Paired domination in trees: A linear algorithm and asymptotic normality

TL;DR

This paper introduces a linear-time algorithm for calculating the paired domination number in trees and demonstrates its asymptotic normality in random rooted trees, including Cayley trees, with specific expectation estimates.

Contribution

It provides the first linear algorithm for paired domination in trees and proves asymptotic normality of the paired domination number in random tree models.

Findings

Linear algorithm for paired domination number in trees

Asymptotic normality of paired domination number in random trees

Expected paired domination number in Cayley trees approaches 0.5177n

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, $\gamma_{\mathrm{pr}}(G)$, 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 $(0.5177\ldots)n$.