28,004 research outputs found
Domination number of graphs with minimum degree five
We prove that for every graph on vertices and with minimum degree
five, the domination number cannot exceed . The proof combines
an algorithmic approach and the discharging method. Using the same technique,
we provide a shorter proof for the known upper bound on the domination
number of graphs of minimum degree four.Comment: 17 page
Tropical Dominating Sets in Vertex-Coloured Graphs
Given a vertex-coloured graph, a dominating set is said to be tropical if
every colour of the graph appears at least once in the set. Here, we study
minimum tropical dominating sets from structural and algorithmic points of
view. First, we prove that the tropical dominating set problem is NP-complete
even when restricted to a simple path. Then, we establish upper bounds related
to various parameters of the graph such as minimum degree and number of edges.
We also give upper bounds for random graphs. Last, we give approximability and
inapproximability results for general and restricted classes of graphs, and
establish a FPT algorithm for interval graphs.Comment: 19 pages, 4 figure
Distributed Dominating Sets on Grids
This paper presents a distributed algorithm for finding near optimal
dominating sets on grids. The basis for this algorithm is an existing
centralized algorithm that constructs dominating sets on grids. The size of the
dominating set provided by this centralized algorithm is upper-bounded by
for grids and its difference
from the optimal domination number of the grid is upper-bounded by five. Both
the centralized and distributed algorithms are generalized for the -distance
dominating set problem, where all grid vertices are within distance of the
vertices in the dominating set.Comment: 10 pages, 9 figures, accepted in ACC 201
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