On the bondage number of planar and directed graphs

The bondage number b(G)b(G) of a nonempty graph G is defined to be the cardinality of the smallest set E of edges of G   such that the graph G-EG-E has domination number greater than that of G  . In this paper we present a simple, intuitive proof that b(G)⩽min{8,Δ(G)+2}b(G)⩽min{8,Δ(G)+2} for all planar graphs G, give examples of planar graphs with bondage number 6, and bound the bondage number of directed graphs.