Remarks on the bondage number of planar graphs

• Published in: Discrete mathematics Vol. 260; no. 1; pp. 57 - 67
• Main Authors: Fischermann, Miranca, Rautenbach, Dieter, Volkmann, Lutz
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 06.01.2003; Elsevier
• Subjects: Applied sciences; Bondage number; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Graph theory; Information retrieval. Graph; Mathematics; Planar graphs; Sciences and techniques of general use; Theoretical computing; Bondage number; Planar graphs; Edge(graph); Cardinal number; Planar graph; Domination number; Graph degree; Discrete mathematics; Connected graph; Vertex(graph)
• DOI: 10.1016/S0012-365X(02)00449-1
• ISSN: 0012-365X

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.