2,058 research outputs found
Disjunctive Total Domination in Graphs
Let be a graph with no isolated vertex. In this paper, we study a
parameter that is a relaxation of arguably the most important domination
parameter, namely the total domination number, . A set of
vertices in is a disjunctive total dominating set of if every vertex is
adjacent to a vertex of or has at least two vertices in at distance2
from it. The disjunctive total domination number, , is the
minimum cardinality of such a set. We observe that . We prove that if is a connected graph of order, then
and we characterize the extremal graphs. It is
known that if is a connected claw-free graph of order, then and this upper bound is tight for arbitrarily large. We show this
upper bound can be improved significantly for the disjunctive total domination
number. We show that if is a connected claw-free graph of order,
then and we characterize the graphs achieving equality
in this bound.Comment: 23 page
Edge Roman domination on graphs
An edge Roman dominating function of a graph is a function satisfying the condition that every edge with
is adjacent to some edge with . The edge Roman
domination number of , denoted by , is the minimum weight
of an edge Roman dominating function of .
This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad stating that if is a graph of maximum degree
on vertices, then . While the counterexamples having the edge Roman domination numbers
, we prove that is an upper bound for connected graphs. Furthermore, we
provide an upper bound for the edge Roman domination number of -degenerate
graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic
graphs.
In addition, we prove that the edge Roman domination numbers of planar graphs
on vertices is at most , which confirms a conjecture of
Akbari and Qajar. We also show an upper bound for graphs of girth at least five
that is 2-cell embeddable in surfaces of small genus. Finally, we prove an
upper bound for graphs that do not contain as a subdivision, which
generalizes a result of Akbari and Qajar on outerplanar graphs
Upper bounds on the k-forcing number of a graph
Given a simple undirected graph and a positive integer , the
-forcing number of , denoted , is the minimum number of vertices
that need to be initially colored so that all vertices eventually become
colored during the discrete dynamical process described by the following rule.
Starting from an initial set of colored vertices and stopping when all vertices
are colored: if a colored vertex has at most non-colored neighbors, then
each of its non-colored neighbors becomes colored. When , this is
equivalent to the zero forcing number, usually denoted with , a recently
introduced invariant that gives an upper bound on the maximum nullity of a
graph. In this paper, we give several upper bounds on the -forcing number.
Notable among these, we show that if is a graph with order and
maximum degree , then . This simplifies to, for the zero forcing number case
of , . Moreover, when and the graph is -connected, we prove that , which is an improvement when , and
specializes to, for the zero forcing number case, . These results resolve a problem posed by
Meyer about regular bipartite circulant graphs. Finally, we present a
relationship between the -forcing number and the connected -domination
number. As a corollary, we find that the sum of the zero forcing number and
connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Exponential Domination in Subcubic Graphs
As a natural variant of domination in graphs, Dankelmann et al. [Domination
with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce
exponential domination, where vertices are considered to have some dominating
power that decreases exponentially with the distance, and the dominated
vertices have to accumulate a sufficient amount of this power emanating from
the dominating vertices. More precisely, if is a set of vertices of a graph
, then is an exponential dominating set of if for every vertex
in , where is the distance
between and in the graph . The exponential domination number of is the minimum
order of an exponential dominating set of .
In the present paper we study exponential domination in subcubic graphs. Our
results are as follows: If is a connected subcubic graph of order ,
then For every , there is some such that
for every cubic graph of girth at least
. For every , there are infinitely many cubic
graphs with . If is a
subcubic tree, then For a given subcubic
tree, can be determined in polynomial time. The minimum
exponential dominating set problem is APX-hard for subcubic graphs
Rainbow Connection Number and Connected Dominating Sets
Rainbow connection number rc(G) of a connected graph G is the minimum number
of colours needed to colour the edges of G, so that every pair of vertices is
connected by at least one path in which no two edges are coloured the same. In
this paper we show that for every connected graph G, with minimum degree at
least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2,
where {\gamma}_c(G) is the connected domination number of G. Bounds of the form
diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special
graph classes follow as easy corollaries from this result. This includes
interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and
chain graphs all with minimum degree at least 2 and connected. We also show
that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of
these cases, we also demonstrate the tightness of the bounds. An extension of
this idea to two-step dominating sets is used to show that for every connected
graph on n vertices with minimum degree {\delta}, the rainbow connection number
is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of
Schiermeyer (2009), improving the previously best known bound of 20n/{\delta}
by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up
to additive factors by a construction of Caro et al. (2008).Comment: 14 page
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