4,589 research outputs found
Complexity and Algorithms for Semipaired Domination in Graphs
For a graph with no isolated vertices, a set is
called a semipaired dominating set of G if is a dominating set of
, and can be partitioned into two element subsets such that the
vertices in each two element set are at distance at most two. The minimum
cardinality of a semipaired dominating set of is called the semipaired
domination number of , and is denoted by . The
\textsc{Minimum Semipaired Domination} problem is to find a semipaired
dominating set of of cardinality . In this paper, we
initiate the algorithmic study of the \textsc{Minimum Semipaired Domination}
problem. We show that the decision version of the \textsc{Minimum Semipaired
Domination} problem is NP-complete for bipartite graphs and split graphs. On
the positive side, we present a linear-time algorithm to compute a minimum
cardinality semipaired dominating set of interval graphs and trees. We also
propose a -approximation algorithm for the \textsc{Minimum
Semipaired Domination} problem, where denote the maximum degree of the
graph and show that the \textsc{Minimum Semipaired Domination} problem cannot
be approximated within for any unless NP
DTIME.Comment: arXiv admin note: text overlap with arXiv:1711.1089
Upper k-tuple total domination in graphs
Let be a simple graph. For any integer , a subset of
is called a -tuple total dominating set of if every vertex in has at
least neighbors in the set. The minimum cardinality of a minimal -tuple
total dominating set of is called the -tuple total domination number of
. In this paper, we introduce the concept of upper -tuple total
domination number of as the maximum cardinality of a minimal -tuple
total dominating set of , and study the problem of finding a minimal
-tuple total dominating set of maximum cardinality on several classes of
graphs, as well as finding general bounds and characterizations. Also, we find
some results on the upper -tuple total domination number of the Cartesian
and cross product graphs
Exact algorithms for dominating induced matchings
Say that an edge of a graph G dominates itself and every other edge adjacent
to it. An edge dominating set of a graph G = (V,E) is a subset of edges E' of E
which dominates all edges of G. In particular, if every edge of G is dominated
by exactly one edge of E' then E' is a dominating induced matching. It is known
that not every graph admits a dominating induced matching, while the problem to
decide if it does admit is NP-complete. In this paper we consider the problem
of finding a minimum weighted dominating induced matching, if any, of a graph
with weighted edges. We describe two exact algorithms for general graphs. The
algorithms are efficient in the cases where G admits a known vertex dominating
set of small size, or when G contains a polynomial number of maximal
independent sets.Comment: 9 page
(Total) Domination in Prisms
With the aid of hypergraph transversals it is proved that , where and denote the total
domination number and the domination number of , respectively, and is
the -dimensional hypercube. More generally, it is shown that if is a
bipartite graph, then . Further, we show
that the bipartite condition is essential by constructing, for any , a
(non-bipartite) graph such that . Along the way several domination-type identities for hypercubes are also
obtained
An time algorithm for minimum weighted dominating induced matching
Say that an edge of a graph dominates itself and every other edge
adjacent to it. An edge dominating set of a graph is a subset of
edges which dominates all edges of . In particular, if
every edge of is dominated by exactly one edge of then is a
dominating induced matching. It is known that not every graph admits a
dominating induced matching, while the problem to decide if it does admit it is
NP-complete. In this paper we consider the problems of finding a minimum
weighted dominating induced matching, if any, and counting the number of
dominating induced matchings of a graph with weighted edges. We describe an
exact algorithm for general graphs that runs in time and
polynomial (linear) space. This improves over any existing exact algorithm for
the problems in consideration.Comment: 17 page
A Note on Integer Domination of Cartesian Product Graphs
Given a graph , a dominating set is a set of vertices such that any
vertex in has at least one neighbor (or possibly itself) in . A
-dominating multiset is a multiset of vertices such that any vertex
in has at least vertices from its closed neighborhood in when
counted with multiplicity. In this paper, we utilize the approach developed by
Clark and Suen (2000) and properties of binary matrices to prove a
"Vizing-like" inequality on minimum -dominating multisets of graphs
and the Cartesian product graph . Specifically, denoting the size of
a minimum -dominating multiset as , we demonstrate that
Vertices in all minimum paired-dominating sets of block graphs
Let be a simple graph without isolated vertices. A set is a paired-dominating set if every vertex in has at least one
neighbor in and the subgraph induced by contains a perfect matching. In
this paper, we present a linear-time algorithm to determine whether a given
vertex in a block graph is contained in all its minimum paired-dominating sets
On Bondage Numbers of Graphs -- a survey with some comments
The bondage number of a nonempty graph is the cardinality of a smallest
edge set whose removal from results in a graph with domination number
greater than the domination number of . This lecture gives a survey on the
bondage number, including the known results, problems and conjectures. We also
summarize other types of bondage numbers.Comment: 80 page; 14 figures; 120 reference
Cartesian product graphs and -tuple total domination
A -tuple total dominating set (TDS) of a graph is a set of
vertices in which every vertex in is adjacent to at least vertices in
; the minimum size of a TDS is denoted . We give
a Vizing-like inequality for Cartesian product graphs, namely provided
, where is the packing number. We
also give bounds on in terms of (open) packing
numbers, and consider the extremal case of ,
i.e., the rook's graph, giving a constructive proof of a general formula for
.Comment: 18 page
Disjoint dominating and 2-dominating sets in graphs
A graph is a -graph if it has a pair of disjoint sets
of vertices of such that is a dominating set and is a
2-dominating set of . We provide several characterizations and hardness
results concerning -graphs.Comment: 15 pages, 3 figure
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