Upper bounds for the bondage number of graphs on topological surfaces

TL;DR

This paper establishes new upper bounds for the bondage number of graphs embedded on various topological surfaces, relating it to maximum degree and surface genus, extending previous results for specific surfaces.

Contribution

The paper generalizes existing bounds for planar and toroidal graphs to graphs on arbitrary orientable and non-orientable surfaces, linking bondage number to surface genus and maximum degree.

Findings

Bound b(G) ≤ Δ(G) + h + 2 for orientable surface of genus h.

Bound b(G) ≤ Δ(G) + k + 1 for non-orientable surface of genus k.

Generalizes known bounds for planar and toroidal graphs.

Abstract

The bondage number b(G) of a graph G is the smallest number of edges of G whose removal from G results in a graph having the domination number larger than that of G. We show that, for a graph G having the maximum vertex degree $\Delta(G)$ and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, $b(G)\le \min\{\Delta(G)+h+2, \Delta(G)+k+1\}$. This generalizes known upper bounds for planar and toroidal graphs.