2 research outputs found
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