6 research outputs found
A Note on Total and Paired Domination of Cartesian Product Graphs
A dominating set for a graph is a subset of such that any
vertex not in has at least one neighbor in . The domination number
is the size of a minimum dominating set in . Vizing's conjecture
from 1968 states that for the Cartesian product of graphs and ,
, and Clark and Suen (2000) proved
that . In this paper, we modify the
approach of Clark and Suen to prove a variety of similar bounds related to
total and paired domination, and also extend these bounds to the -Cartesian
product of graphs through
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
(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
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
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
-tuple total domination number of rook's graphs
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 called the -tuple total dominating
number and it is denoted by . We give a constructive
proof of a general formula for