The bondage number b(G) of a nonempty graph G is the cardinality of a smallest edge set whose removal from G results in a graph with domination number greater than the domination number (G) ofG. Kang andYuan proved b(G) 8 for every connected planar graph G. Fischermann, Rautenbach and Volkmann obtained some further results for connected planar graphs. In this paper, we generalize their results ...

A virtual link may be defined as an equivalence class of diagrams, or alternatively a stable links in thickened surfaces. We prove that minimal crossing diagram has genus across representatives the class. This is achieved by constructing new parity theory for links. As corollaries, we crossing, bridge, and ascending numbers classical do not decrease when it regarded link. extends corresponding ...