60,264 research outputs found
Bounds on the 2-domination number
In a graph G, a set D⊆V(G) is called 2-dominating set if each vertex not in D has at least two neighbors in D. The 2-domination number γ2(G) is the minimum cardinality of such a set D. We give a method for the construction of 2-dominating sets, which also yields upper bounds on the 2-domination number in terms of the number of vertices, if the minimum degree δ(G) is fixed. These improve the best earlier bounds for any 6≤δ(G)≤21. In particular, we prove that γ2(G) is strictly smaller than n/2, if δ(G)≥6. Our proof technique uses a weight-assignment to the vertices where the weights are changed during the procedure. © 2017 Elsevier B.V
Coloring, location and domination of corona graphs
A vertex coloring of a graph is an assignment of colors to the vertices
of such that every two adjacent vertices of have different colors. A
coloring related property of a graphs is also an assignment of colors or labels
to the vertices of a graph, in which the process of labeling is done according
to an extra condition. A set of vertices of a graph is a dominating set
in if every vertex outside of is adjacent to at least one vertex
belonging to . A domination parameter of is related to those structures
of a graph satisfying some domination property together with other conditions
on the vertices of . In this article we study several mathematical
properties related to coloring, domination and location of corona graphs.
We investigate the distance- colorings of corona graphs. Particularly, we
obtain tight bounds for the distance-2 chromatic number and distance-3
chromatic number of corona graphs, throughout some relationships between the
distance- chromatic number of corona graphs and the distance- chromatic
number of its factors. Moreover, we give the exact value of the distance-
chromatic number of the corona of a path and an arbitrary graph. On the other
hand, we obtain bounds for the Roman dominating number and the
locating-domination number of corona graphs. We give closed formulaes for the
-domination number, the distance- domination number, the independence
domination number, the domatic number and the idomatic number of corona graphs.Comment: 18 page
A Greedy Partition Lemma for Directed Domination
A directed dominating set in a directed graph is a set of vertices of
such that every vertex has an adjacent vertex
in with directed to . The directed domination number of , denoted
by , is the minimum cardinality of a directed dominating set in .
The directed domination number of a graph , denoted , which is
the maximum directed domination number over all orientations of
. The directed domination number of a complete graph was first studied by
Erd\"{o}s [Math. Gaz. 47 (1963), 220--222], albeit in disguised form. In this
paper we prove a Greedy Partition Lemma for directed domination in oriented
graphs. Applying this lemma, we obtain bounds on the directed domination
number. In particular, if denotes the independence number of a graph
, we show that .Comment: 12 page
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
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