694 research outputs found
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
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
Weak and Strong Reinforcement Number For a Graph
Introducing the weak reinforcement number which is the minimum number of added edges to reduce the weak dominating number, and giving some boundary of this new parameter and trees
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
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
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