6,192 research outputs found
The bondage number of graphs on topological surfaces and Teschner's conjecture
The bondage number of a graph is the smallest number of its edges whose
removal results in a graph having a larger domination number. We provide
constant upper bounds for the bondage number of graphs on topological surfaces,
improve upper bounds for the bondage number in terms of the maximum vertex
degree and the orientable and non-orientable genera of the graph, and show
tight lower bounds for the number of vertices of graphs 2-cell embeddable on
topological surfaces of a given genus. Also, we provide stronger upper bounds
for graphs with no triangles and graphs with the number of vertices larger than
a certain threshold in terms of the graph genera. This settles Teschner's
Conjecture in positive for almost all graphs.Comment: 21 pages; Original version from January 201
On the Roman Bondage Number of Graphs on surfaces
A Roman dominating function on a graph is a labeling such that every vertex with label has a neighbor
with label . The Roman domination number, , of is the
minimum of over such functions. The Roman bondage
number is the cardinality of a smallest set of edges whose removal
from results in a graph with Roman domination number not equal to
. In this paper we obtain upper bounds on in terms of
(a) the average degree and maximum degree, and (b) Euler characteristic, girth
and maximum degree. We also show that the Roman bondage number of every graph
which admits a -cell embedding on a surface with non negative Euler
characteristic does not exceed .Comment: 5 page
Graph Configurations and Independent Bondage Numbers of Planar Graphs
The independent domination number of a finite graph G is the minimum
cardinality of an independent dominating set of vertices. The independent
bondage number of G is the minimum cardinality of a set of edges whose deletion
results in a graph with a larger independent domination number than that of G.
In this research, we enhance the existing upper bound on the independent
bondage number of a planar graph with a minimum degree of at least three by
identifying specific configurations within such planar graphs
Upper bounds for the bondage number of graphs on topological surfaces
The bondage number b(G) of a graph G is the smallest number of edges of G
whose removal from G results in a graph having the domination number larger
than that of G. We show that, for a graph G having the maximum vertex degree
and embeddable on an orientable surface of genus h and a
non-orientable surface of genus k, . This generalizes known upper bounds for planar and toroidal
graphs.Comment: 10 pages; Updated version (April 2011); Presented at the 7th ECCC,
Wolfville (Nova Scotia, Canada), May 4-6, 2011, and the 23rd BCC, Exeter
(England, UK), July 3-8, 201
Upper bounds for domination related parameters in graphs on surfaces
AbstractIn this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum degree
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