47 research outputs found
Bounds On Fuzzy Dominator Chromatic Number of Fuzzy Soft Bipartite Graphs
An FSG GS(T,V) fuzzy’s soft dominator colouring (FSDC) is a suitable Fuzzy Soft Colouring (FSC) where every node of a colour groupis dominated by a vertex of GS(T,V). In the current work, we characterize the sharp bounds for the Fuzzy Dominator Chromatic Number(FDCN) of fuzzy soft bipartite graphs and we present limits on theFDCN of fuzzy soft bipartite graph in terms of the γe(GS(T; V )).Furthermore, we classify fuzzy soft bipartite graphs into three classesbased on FDC
Kernelization and Sparseness: the case of Dominating Set
We prove that for every positive integer and for every graph class
of bounded expansion, the -Dominating Set problem admits a
linear kernel on graphs from . Moreover, when is only
assumed to be nowhere dense, then we give an almost linear kernel on for the classic Dominating Set problem, i.e., for the case . These
results generalize a line of previous research on finding linear kernels for
Dominating Set and -Dominating Set. However, the approach taken in this
work, which is based on the theory of sparse graphs, is radically different and
conceptually much simpler than the previous approaches.
We complement our findings by showing that for the closely related Connected
Dominating Set problem, the existence of such kernelization algorithms is
unlikely, even though the problem is known to admit a linear kernel on
-topological-minor-free graphs. Also, we prove that for any somewhere dense
class , there is some for which -Dominating Set is
W[]-hard on . Thus, our results fall short of proving a sharp
dichotomy for the parameterized complexity of -Dominating Set on
subgraph-monotone graph classes: we conjecture that the border of tractability
lies exactly between nowhere dense and somewhere dense graph classes.Comment: v2: new author, added results for r-Dominating Sets in bounded
expansion graph
Dominator Coloring and CD Coloring in Almost Cluster Graphs
In this paper, we study two popular variants of Graph Coloring -- Dominator
Coloring and CD Coloring. In both problems, we are given a graph and a
natural number as input and the goal is to properly color the vertices
with at most colors with specific constraints. In Dominator Coloring, we
require for each , a color such that dominates all vertices
colored . In CD Coloring, we require for each color , a
which dominates all vertices colored . These problems, defined due to their
applications in social and genetic networks, have been studied extensively in
the last 15 years. While it is known that both problems are fixed-parameter
tractable (FPT) when parameterized by where is the treewidth of
, we consider strictly structural parameterizations which naturally arise
out of the problems' applications.
We prove that Dominator Coloring is FPT when parameterized by the size of a
graph's cluster vertex deletion (CVD) set and that CD Coloring is FPT
parameterized by CVD set size plus the number of remaining cliques. En route,
we design a simpler and faster FPT algorithms when the problems are
parameterized by the size of a graph's twin cover, a special CVD set. When the
parameter is the size of a graph's clique modulator, we design a randomized
single-exponential time algorithm for the problems. These algorithms use an
inclusion-exclusion based polynomial sieving technique and add to the growing
number of applications using this powerful algebraic technique.Comment: 29 pages, 3 figure
On dominator colorings in graphs
The article of record as published may be located at http://gtn.kazlow.info/GTN54.pdfGraph Theory Notes of New York LII, (2007) 25-30Given a graph G, the dominator coloring problem seeks a proper coloring of G with the additional property that every vertex in the graph dominates an entire color class. We seek to minimize the number of color classes. We study this problem on several classes of graphs, as well as finding general bounds and characterizations. We also show the relation between dominator chromatic number, chromatic number, and domination number