536 research outputs found
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
Data Reduction for Graph Coloring Problems
This paper studies the kernelization complexity of graph coloring problems
with respect to certain structural parameterizations of the input instances. We
are interested in how well polynomial-time data reduction can provably shrink
instances of coloring problems, in terms of the chosen parameter. It is well
known that deciding 3-colorability is already NP-complete, hence parameterizing
by the requested number of colors is not fruitful. Instead, we pick up on a
research thread initiated by Cai (DAM, 2003) who studied coloring problems
parameterized by the modification distance of the input graph to a graph class
on which coloring is polynomial-time solvable; for example parameterizing by
the number k of vertex-deletions needed to make the graph chordal. We obtain
various upper and lower bounds for kernels of such parameterizations of
q-Coloring, complementing Cai's study of the time complexity with respect to
these parameters.
Our results show that the existence of polynomial kernels for q-Coloring
parameterized by the vertex-deletion distance to a graph class F is strongly
related to the existence of a function f(q) which bounds the number of vertices
which are needed to preserve the NO-answer to an instance of q-List-Coloring on
F.Comment: Author-accepted manuscript of the article that will appear in the FCT
2011 special issue of Information & Computatio
Limited packings of closed neighbourhoods in graphs
The k-limited packing number, , of a graph , introduced by
Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set
of vertices of such that every vertex of has at most elements of
in its closed neighbourhood. The main aim in this paper is to prove the
best-possible result that if is a cubic graph, then , improving the previous lower bound given by Gallant, \emph{et al.}
In addition, we construct an infinite family of graphs to show that lower
bounds given by Gagarin and Zverovich are asymptotically best-possible, up to a
constant factor, when is fixed and tends to infinity. For
tending to infinity and tending to infinity sufficiently
quickly, we give an asymptotically best-possible lower bound for ,
improving previous bounds
Resolving sets for breaking symmetries of graphs
This paper deals with the maximum value of the difference between the
determining number and the metric dimension of a graph as a function of its
order. Our technique requires to use locating-dominating sets, and perform an
independent study on other functions related to these sets. Thus, we obtain
lower and upper bounds on all these functions by means of very diverse tools.
Among them are some adequate constructions of graphs, a variant of a classical
result in graph domination and a polynomial time algorithm that produces both
distinguishing sets and determining sets. Further, we consider specific
families of graphs where the restrictions of these functions can be computed.
To this end, we utilize two well-known objects in graph theory: -dominating
sets and matchings.Comment: 24 pages, 12 figure
Advances in Discrete Applied Mathematics and Graph Theory
The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs
- …