Paired-domination in binary trees

Abstract

A set $S$ of vertices in an isolate-free graph $G$ is a paired-dominating set if every vertex of $G$ is adjacent to some other vertex in $S$ and the subgraph $G\left[S\right]$ induced by the set $S$ contains a perfect matching. The paired-domination number ${\gamma }_{\mathrm{pr}}\left(G\right)$ of $G$ is the minimum cardinality of a paired-dominating set in $G$. A binary tree is a tree in which every vertex has degree 1 or degree 3. We determine a tight upper bound on the paired-domination number of a binary tree and show that if $T$ is a binary tree of order $n\ge 4$, then ${\gamma }_{\mathrm{pr}}\left(T\right)\le \frac{2}{3}(n-1)$. Thereafter we continue the study of a version of the paired-domination game recently introduced by the authors (Gray and Henning, 2023) that embraces both the domination and matching flavor of the game. We give an explicit formula for the game paired-domination number of an infinite family of binary trees and show that if $T$ is a binary caterpillar of order $n$, then ${\gamma }_{\mathrm{gpr}}\left(T\right)=\frac{3}{4}n-\Phi \left(n\right)$, where $\Phi \left(n\right)$ takes on one of the values in the set $\{1,\frac{3}{2},2,\frac{5}{2}\}$. We show that if $T$ is a complete binary tree of order $n$, then ${\gamma }_{\mathrm{gpr}}\left(T\right)<(\frac{2}{3}+\frac{1}{192})n$. We conclude with a conjecture that $sup\frac{{\gamma }_{\mathrm{gpr}}\left(T\right)}{n}=\frac{3}{4}$ where the supremum is over all binary trees $T$ of order $n\ge 4$.