1,292 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
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
The bondage number of random graphs
A dominating set of a graph is a subset of its vertices such that every
vertex not in is adjacent to at least one member of . The domination
number of a graph is the number of vertices in a smallest dominating set of
. The bondage number of a nonempty graph is the size of a smallest set
of edges whose removal from results in a graph with domination number
greater than the domination number of . In this note, we study the bondage
number of binomial random graph . We obtain a lower bound that matches
the order of the trivial upper bound. As a side product, we give a one-point
concentration result for the domination number of under certain
restrictions
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
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
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
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
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