The Bondage Number of Graphs with Crossing Number Less than Four

The bondage number b(G) of a graph G is the smallest number of edges whose removal results in a graph with domination number greater than the domination number of G. Kang and Yuan [Bondage number of planar graphs. Discrete Math. 222 (2000), 191198] proved b(G) 6 min{8,∆+ 2} for every connected planar graph G, where ∆ is the maximum degree of G. Later Carlson and Develin [On the bondage number of planar and directed graphs. Discrete Math. 306 (8-9) (2006), 820-826] presented a method to give a short proof for this result. This paper applies this technique to generalize the result of Kang and Yuan to any connected graph with crossing number less than four.