17,388 research outputs found
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
Hybridizing Non-dominated Sorting Algorithms: Divide-and-Conquer Meets Best Order Sort
Many production-grade algorithms benefit from combining an asymptotically
efficient algorithm for solving big problem instances, by splitting them into
smaller ones, and an asymptotically inefficient algorithm with a very small
implementation constant for solving small subproblems. A well-known example is
stable sorting, where mergesort is often combined with insertion sort to
achieve a constant but noticeable speed-up.
We apply this idea to non-dominated sorting. Namely, we combine the
divide-and-conquer algorithm, which has the currently best known asymptotic
runtime of , with the Best Order Sort algorithm, which
has the runtime of but demonstrates the best practical performance
out of quadratic algorithms.
Empirical evaluation shows that the hybrid's running time is typically not
worse than of both original algorithms, while for large numbers of points it
outperforms them by at least 20%. For smaller numbers of objectives, the
speedup can be as large as four times.Comment: A two-page abstract of this paper will appear in the proceedings
companion of the 2017 Genetic and Evolutionary Computation Conference (GECCO
2017
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