Independent Domination of k-Trees

TL;DR

The paper addresses the independent domination number $\gamma_i(G)$ for the class of $k$-trees, a natural generalization of trees and maximal outerplanar graphs. It develops a coloring-based constructive approach by proving the existence of a proper $(k+2)$-coloring in which every $(k+1)^+$-vertex's closed neighborhood contains all colors, enabling independent domination constructions. The main result is the tight bound $\gamma_i(G) \le \left\lfloor \frac{n+|V_k^G|}{k+2} \right\rfloor$, with a tightness demonstration on $k$-paths via a constructed graph $P'$ where $\gamma_i(P')=t$. These results extend known bounds for $1$-trees and maximal outerplanar graphs to all $k$-trees and connect the bound to the structure of the $k$-vertices, offering insight into domination in bounded-treewidth graphs and guiding future work on partial $k$-trees and related invariants.

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 $γ_i(G)$ 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 $k\in \mathbb{N}$.

Figures (3)

• Figure 1: Constructing 2-trees from existing 2-trees
• Figure 2: The caterpillar graph on $2m$ vertices.
• Figure 3: Constructing a 3-tree from the 3-path with $t=4$.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Corollary 1
• Theorem 3
• proof