The bondage number of graphs on topological surfaces: degree-S vertices and the average degree

TL;DR

This paper establishes new upper bounds for the bondage number of graphs embedded on various topological surfaces, relating it to parameters like genus, degree, domination number, and girth, with improved bounds and conditions.

Contribution

It introduces improved upper bounds for the bondage number based on surface genus, degree, and domination number, along with new conditions for these bounds to hold.

Findings

Bound of b(G) ≤ 7 + i for non-orientable surfaces with genus i=1,2,3.

Bound of b(G) ≤ 12 for certain non-orientable and orientable surfaces.

Upper bounds on bondage number depending on degree, Euler characteristic, and domination number.

Abstract

The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. An orientable surface $\mathbb{S}_h$ of genus $h$, $h \geq 0$, is obtained from the sphere $\mathbb{S}_0$ by adding $h$ handles. A non-orientable surface $\mathbb{N}_q$ of genus $q$, $q \geq 1$, is obtained from the sphere by adding $q$ crosscaps. The Euler characteristic of a surface is defined by $\chi(\mathbb{S}_h) = 2 - 2h$ and $\chi(\mathbb{S}_q)= 2-q$. Let $G$ be a connected graph of order $n$ which is 2-cell embedded on a surface $\mathbb{M}$ with $\chi(\mathbb{M})= \chi$. We prove that $b(G) \leq 7+i$ when $\mathbb{M} = \mathbb{N}_i$, $i=1,2,3$, and $b(G) \leq 12$ when $\mathbb{M} \in \{\mathbb{N}_4, \mathbb{S}_2\}$. We give new arguments that improve the known upper bounds on the bondage number at least when $-7\chi/(\delta(G) - 5) < n \leq -12\chi$, $\delta(G) \geq 6$, where $\delta(G)$ is the minimum degree of $G$. We obtain sufficient conditions for the validity of the inequality $b(G) \leq 2s-2$, provided $G$ has degree $s$ vertices. In particular, we prove that if $\delta (G) = \delta \geq 6$, $\chi \leq -1$ and $-14\chi < \delta - 4 + 2(\delta -5)n$ then $b(G) \leq 2\delta -2$. We show that if $\gamma (G) = \gamma \not = 2$, where $\gamma (G)$ is the domination number of $G$, then $n \geq \gamma + (1 + \sqrt{9+8\gamma-8\chi})/2$; the bound is tight. We also present upper bounds for the bondage number of graphs in terms of the girth, domination number and Euler characteristic. As a corollary we prove that if $\gamma(G) \geq 4$ and $\chi \leq -1$, then $b(G) \leq 11 - 24\chi/(9 + \sqrt{41 - 8\chi})$. Several unanswered questions are posed.