3 research outputs found
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
Total Domination, Separated Clusters, CD-Coloring: Algorithms and Hardness
Domination and coloring are two classic problems in graph theory. The major
focus of this paper is the CD-COLORING problem which combines the flavours of
domination and colouring. Let be an undirected graph. A proper vertex
coloring of is a if each color class has a dominating vertex
in . The minimum integer for which there exists a of
using colors is called the cd-chromatic number, . A set
is a total dominating set if any vertex in has a neighbor
in . The total domination number, of is the minimum
integer such that has a total dominating set of size . A set
is a if no two vertices in lie at a
distance 2 in . The separated-cluster number, , of is the
maximum integer such that has a separated-cluster of size .
In this paper, first we explore the connection between CD-COLORING and TOTAL
DOMINATION. We prove that CD-COLORING and TOTAL DOMINATION are NP-Complete on
triangle-free -regular graphs for each fixed integer . We also
study the relationship between the parameters and .
Analogous to the well-known notion of `perfectness', here we introduce the
notion of `cd-perfectness'. We prove a sufficient condition for a graph to
be cd-perfect (i.e. , for any induced subgraph
of ) which is also necessary for certain graph classes (like triangle-free
graphs). Here, we propose a generalized framework via which we obtain several
exciting consequences in the algorithmic complexities of special graph classes.
In addition, we settle an open problem by showing that the SEPARATED-CLUSTER is
polynomially solvable for interval graphs