Independent Bondage Number in Graphs under Girth Constraints

TL;DR

This paper investigates the independent bondage number in graphs, especially planar graphs with girth constraints, providing new upper bounds and extending previous results on bondage numbers.

Contribution

It establishes new upper bounds on the independent bondage number for planar graphs with specific girth and minimum degree conditions, extending prior work.

Findings

Upper bounds for planar graphs with girth ≥ 5 and δ(G) ≥ 2

Bounds for graphs with girth ≥ 4 and δ(G) ≥ 3

Results for graphs with girth ≥ 10 and δ(G) ≥ 2

Abstract

Given a finite, simple graph $G$, the independent bondage number of $G$ is the minimum size of an edge set such that its deletion results in a graph with strictly larger independent domination number than that of $G$. While the bondage number of graphs under girth constraints has been studied, very few results have yet been established for the independent bondage number. In this study, we establish upper bounds on the independent bondage number of planar graphs under given girth constraints, extending results on the bondage number by Fischermann, Rautenbach, and Volkmann and on the structures of planar graphs by Borodin and Ivanova. In particular, we identify additional structures and establish bounds on the independent bondage number for planar graphs with $\delta (G) \geq 2$ and $g(G)\geq 5$, $\delta(G)\geq 3$ and $g(G)\geq 4$, and $\delta (G) \geq 2$ and $g(G)\geq 10$.