4 research outputs found
An improved upper bound for the bondage number of graphs on surfaces
The bondage number of a graph is the smallest number of edges
whose removal from results in a graph with larger domination number.
Recently Gagarin and Zverovich showed that, for a graph with maximum degree
and embeddable on an orientable surface of genus and a
non-orientable surface of genus ,
. They also gave examples showing
that adjustments of their proofs implicitly provide better results for larger
values of and . In this paper we establish an improved explicit upper
bound for , using the Euler characteristic instead of the genera
and , with the relations and . We show that
for the case (i.e. or
), where is the largest real root of the cubic equation
. Our proof is based on the technique
developed by Carlson-Develin and Gagarin-Zverovich, and includes some
elementary calculus as a new ingredient. We also find an asymptotically
equivalent result for
, and a further improvement for graphs with large girth.Comment: 8 pages, to appear in Discrete Mathematic