118 research outputs found
The total bondage number of grid graphs
The total domination number of a graph without isolated vertices is the
minimum number of vertices that dominate all vertices in . The total bondage
number of is the minimum number of edges whose removal enlarges
the total domination number. This paper considers grid graphs. An -grid
graph is defined as the cartesian product of two paths and
. This paper determines the exact values of and
, and establishes some upper bounds of .Comment: 15 pages with 4 figure
Bondage number of grid graphs
The bondage number of a nonempty graph is the cardinality of a
smallest set of edges whose removal from results in a graph with domination
number greater than the domination number of . Here we study the bondage
number of some grid-like graphs. In this sense, we obtain some bounds or exact
values of the bondage number of some strong product and direct product of two
paths.Comment: 13 pages. Discrete Applied Mathematics, 201
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
A Study on Set-Graphs
A \textit{primitive hole} of a graph is a cycle of length in . The
number of primitive holes in a given graph is called the primitive hole
number of that graph . The primitive degree of a vertex of a given graph
is the number of primitive holes incident on the vertex . In this paper,
we introduce the notion of set-graphs and study the properties and
characteristics of set-graphs. We also check the primitive hole number and
primitive degree of set-graphs. Interesting introductory results on the nature
of order of set-graphs, degree of the vertices corresponding to subsets of
equal cardinality, the number of largest complete subgraphs in a set-graph etc.
are discussed in this study. A recursive formula to determine the primitive
hole number of a set-graph is also derived in this paper.Comment: 11 pages, 1 figure, submitte
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
A bound on the size of a graph with given order and bondage number
AbstractThe domination number of a graph is the minimum number of vertices in a set S such that every vertex of the graph is either in S or adjacent to a member of S. The bondage number of a graph G is the cardinality of a smallest set of edges whose removal results in a graph with domination number greater than that of G. We prove that a graph with p vertices and bondage number b has at least p(b + 1)/4 edges, and for each b there is at least one p for which this bound is sharp. © 1999 Elsevier Science B.V. All rights reserve
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