The bondage number of graphs on topological surfaces and Teschners conjecture

The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of the graph, and show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus. Also, we provide stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of the graph genera. This settles Teschner’s Conjecture in positive for almost all graphs.

💡 Research Summary

The paper investigates the bondage number b(G) of a graph G, defined as the smallest number of edges whose removal strictly increases the domination number γ(G). While previous work has offered general bounds such as b(G) ≤ Δ(G)+2 (where Δ is the maximum degree), these bounds ignore the topological constraints that arise when a graph is embedded on a surface. The authors focus on graphs that admit a 2‑cell embedding on an orientable surface of genus g or a non‑orientable surface of genus γ, and they derive a series of refined upper and lower bounds that incorporate the surface genus, the maximum degree, and the presence or absence of triangles.

Main contributions

1. Genus‑dependent constant upper bound.
   By combining Euler’s formula (V − E + F = 2 − 2g for orientable surfaces, 2 − γ for non‑orientable) with degree‑sum arguments, the authors prove
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