15 research outputs found
Nordhaus-Gaddum for Treewidth
We prove that for every graph with vertices, the treewidth of
plus the treewidth of the complement of is at least . This bound is
tight
Nordhaus-Gaddum bounds for locating domination
A dominating set S of graph G is called metric-locating-dominating if it is
also locating, that is, if every vertex v is uniquely determined by its vector
of distances to the vertices in S. If moreover, every vertex v not in S is also
uniquely determined by the set of neighbors of v belonging to S, then it is
said to be locating-dominating. Locating, metric-locating-dominating and
locating-dominating sets of minimum cardinality are called b-codes, e-codes and
l-codes, respectively. A Nordhaus-Gaddum bound is a tight lower or upper bound
on the sum or product of a parameter of a graph G and its complement G. In this
paper, we present some Nordhaus-Gaddum bounds for the location number b, the
metric-location-number e and the location-domination number l. Moreover, in
each case, the graph family attaining the corresponding bound is characterized.Comment: 7 pages, 2 figure
An updated survey on rainbow connections of graphs - a dynamic survey
The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow -connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study
Complementary Vanishing Graphs
Given a graph with vertices , we define
to be the set of symmetric matrices such that
for we have if and only if . Motivated
by the Graph Complement Conjecture, we say that a graph is complementary
vanishing if there exist matrices and such that . We provide combinatorial
conditions for when a graph is or is not complementary vanishing, and we
characterize which graphs are complementary vanishing in terms of certain
minimal complementary vanishing graphs. In addition to this, we determine which
graphs on at most vertices are complementary vanishing
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
Study of the Gromov hyperbolicity constant on graphs
The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space and Riemannian manifolds of negative sectional curvature. It is remarkable that a simple concept leads to such a rich general theory. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of any geodesic metric space is equivalent to the hyperbolicity of a graph related to it.
In this Ph. D. Thesis we characterize the hyperbolicity constant of interval graphs and circular-arc graphs. Likewise, we provide relationships between dominant sets and the hyperbolicity constant.
Finally, we study the invariance of the hyperbolicity constant when the graphs are transformed by several operators.Programa de Doctorado en IngenierÃa Matemática por la Universidad Carlos III de MadridPresidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva TourÃs Loj
A Linear Kernel for Planar Total Dominating Set
A total dominating set of a graph is a subset such
that every vertex in is adjacent to some vertex in . Finding a total
dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on
general graphs when parameterized by the solution size. By the meta-theorem of
Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total
Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how
such a kernel can be effectively constructed, and how to obtain explicit
reduction rules with reasonably small constants. Following the approach of
Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating
Set on planar graphs with at most vertices, where is the size of the
solution. This result complements several known constructive linear kernels on
planar graphs for other domination problems such as Dominating Set, Edge
Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue
Dominating Set.Comment: 33 pages, 13 figure