Remarks on the bondage number of planar graphs

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest set of edges whose removal from G results in a graph with domination number greater than the domination number γ(G) of G. In 1998, J.E. Dunbar, T.W. Haynes, U. Teschner, and L. Volkmann posed the conjecture b(G) ≤ Δ(G) + 1 for every nontrivial connected planar graph G. Two years later, L. Kang and J. Yuan proved b(G) ≤ 8 for every connected planar graph G, and therefore, they confirmed the conjecture for Δ(G) ≥ 7. In this paper we show that this conjecture is valid for all connected planar graphs of girth g(G) ≥ 4 and maximum degree Δ(G) ≥ 5 as well as for all not 3-regular graphs of girth g(G) ≥ 5. Some further related results and open problems are also presented.