Independent Bondage Number in Graphs under Girth Constraints
E. G. K. M. Gamlath, Andrew Pham, Bing Wei
TL;DR
This work addresses the independent bondage number $b_i(G)$ for planar graphs under girth constraints. It employs the discharging method to derive constant upper bounds on $b_i(G)$ across regimes defined by minimum degree and girth: $b_i(G)\le 5$ for $\nabla(G)\ge 2$, $g(G)\ge 5$; $b_i(G)\le 6$ for $\nabla(G)\ge 3$, $g(G)\ge 4$; $b_i(G)\le 4$ for $\nabla(G)\ge 2$, $g(G)\ge 7$; and $b_i(G)\le 3$ for $\nabla(G)\ge 2$, $g(G)\ge 10$. The results extend prior bondage studies and showcase how planar-graph structure constrains the independent bondage number. The paper provides a new, systematic set of configurations and discharging arguments that yield tight, yet constant, upper bounds, contributing to a clearer understanding of network vulnerability under girth constraints. The findings have potential implications for planarity-based network design and resilience analysis where independent domination plays a key role.
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 $δ(G) \geq 2$ and $g(G)\geq 5$, $δ(G)\geq 3$ and $g(G)\geq 4$, and $δ(G) \geq 2$ and $g(G)\geq 10$.