Remarks on the bondage number of planar graphs (Q1861243)

A subset \(S\) of the vertex set of a graph \(G\) is called dominating in \(G\), if each vertex of \(G\) either is in \(S\), or is adjacent to a vertex of \(S\). The minimum number of vertices of a dominating set in \(G\) is the domination number \(\gamma(G)\) of \(G\). The minimum number of vertices whose deletion from \(G\) changes the value of \(\gamma(G)\) for a higher one, is the bondage number \(b(G)\) of \(G\). This work extends the conjecture that \(b(G) \leq \Delta (G)\) for every connected planar graph to all connected planar graph of girth \(g(G) \geq 4\) and maximum degree at least 5.