1,762 research outputs found
k-Tuple_Total_Domination_in_Inflated_Graphs
The inflated graph of a graph with vertices is obtained
from by replacing every vertex of degree of by a clique, which is
isomorph to the complete graph , and each edge of is
replaced by an edge in such a way that , , and
two different edges of are replaced by non-adjacent edges of . For
integer , the -tuple total domination number of is the minimum cardinality of a -tuple total dominating set
of , which is a set of vertices in such that every vertex of is
adjacent to at least vertices in it. For existing this number, must the
minimum degree of is at least . Here, we study the -tuple total
domination number in inflated graphs when . First we prove that
, and then we
characterize graphs that the -tuple total domination number number of
is or . Then we find bounds for this number in the
inflated graph , when has a cut-edge or cut-vertex , in terms
on the -tuple total domination number of the inflated graphs of the
components of or -components of , respectively. Finally, we
calculate this number in the inflated graphs that have obtained by some of the
known graphs
{Linear Kernels for -Tupel and Liar's Domination in Bounded Genus Graphs}
A set is called a -tuple dominating set of a graph if for all , where denotes the closed neighborhood of . A set is called a liar's dominating set of a graph if (i) for all and (ii) for every pair of distinct vertices , . Given a graph , the decision versions of -Tuple Domination Problem and the Liar's Domination Problem are to check whether there exists a -tuple dominating set and a liar's dominating set of of a given cardinality, respectively. These two problems are known to be NP-complete \cite{LiaoChang2003, Slater2009}. In this paper, we study the parameterized complexity of these problems. We show that the -Tuple Domination Problem and the Liar's Domination Problem are -hard for general graphs but they admit linear kernels for graphs with bounded genus.Title changed from "Parameterized complexity of k-tuple and liar's domination" to "Linear kernels for k-tuple and liar's domination in bounded genus graphs
Upper bounds for alpha-domination parameters
In this paper, we provide a new upper bound for the alpha-domination number.
This result generalises the well-known Caro-Roditty bound for the domination
number of a graph. The same probabilistic construction is used to generalise
another well-known upper bound for the classical domination in graphs. We also
prove similar upper bounds for the alpha-rate domination number, which combines
the concepts of alpha-domination and k-tuple domination.Comment: 7 pages; Presented at the 4th East Coast Combinatorial Conference,
Antigonish (Nova Scotia, Canada), May 1-2, 200
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