Independent Domination of k-Trees

TL;DR

This paper establishes a tight upper bound for the independent domination number of k-trees, extending previous results from 1-trees and outerplanar graphs to all k-trees, a broad class of graphs.

Contribution

It generalizes existing bounds on independent domination numbers from specific graph subclasses to all k-trees, providing a comprehensive theoretical result.

Findings

Derived a tight upper bound for the independent domination number of k-trees.

Extended known results from 1-trees and outerplanar graphs to all k-trees.

Contributed to the theoretical understanding of domination parameters in graph theory.

Abstract

Given a simple, finite, nonempty graph $G=(V(G),E(G))$, a vertex subset $D\subseteq V(G)$ is said to be a dominating set if every vertex $v\in V(G)-D$ is adjacent to a vertex in $D$. The independent domination number $\gamma_i(G)$ is the minimum cardinality among all independent dominating sets of $G$. Since determining the domination number for general graphs is NP-complete, we focus on the class of $k$-trees. Favaron established a tight upper bound for $1$-trees, while Campos and Wakabayashi determined a tight upper bound for maximal outerplanar graphs, a subclass of $2$-trees. We generalize these results and establish a tight upper bound for the independent domination number of $k$-trees for all $k\in \mathbb{N}$.