6,339 research outputs found
An Efficient Algorithm for Mixed Domination on Generalized Series-Parallel Graphs
A mixed dominating set of a graph is a subset such that each element is adjacent or
incident to at least one element in . The mixed domination number
of a graph is the minimum cardinality among all mixed
dominating sets in . The problem of finding is know to be
NP-complete. In this paper, we present an explicit polynomial-time algorithm to
construct a mixed dominating set of size by a parse tree when
is a generalized series-parallel graph
On Complexities of Minus Domination
A function f: V \rightarrow \{-1,0,1\} is a minus-domination function of a
graph G=(V,E) if the values over the vertices in each closed neighborhood sum
to a positive number. The weight of f is the sum of f(x) over all vertices x
\in V. The minus-domination number \gamma^{-}(G) is the minimum weight over all
minus-domination functions. The size of a minus domination is the number of
vertices that are assigned 1. In this paper we show that the minus-domination
problem is fixed-parameter tractable for d-degenerate graphs when parameterized
by the size of the minus-dominating set and by d. The minus-domination problem
is polynomial for graphs of bounded rankwidth and for strongly chordal graphs.
It is NP-complete for splitgraphs. Unless P=NP there is no fixed-parameter
algorithm for minus-domination. 79,1 5
Power domination throttling
A power dominating set of a graph is a set that colors
every vertex of according to the following rules: in the first timestep,
every vertex in becomes colored; in each subsequent timestep, every
vertex which is the only non-colored neighbor of some colored vertex becomes
colored. The power domination throttling number of is the minimum sum of
the size of a power dominating set and the number of timesteps it takes
to color the graph. In this paper, we determine the complexity of power
domination throttling and give some tools for computing and bounding the power
domination throttling number. Some of our results apply to very general
variants of throttling and to other aspects of power domination.Comment: 19 page
Layered graphs: a class that admits polynomial time solutions for some hard problems
The independent set on a graph is a subset of such that no two
vertices in the subset have an edge between them. The MIS problem on seeks
to identify an independent set with maximum cardinality, i.e. maximum
independent set or MIS. is a vertex cover if every
edge in the graph is incident upon at least one vertex in . is dominating set of if forall either or
and . A connected dominating set, CDS, is a
dominating set that forms a single component in . The MVC problem on
seeks to identify a vertex cover with minimum cardinality, i.e. minimum vertex
cover or MVC. Likewise, CVC seeks a connected vertex cover (CVC) with minimum
cardinality. The problems MDS and CDS seek to identify a dominating set and a
connected dominating set respectively of minimum cardinalities. MVC, CVC, MDS,
and CDS on a general graph are known to be NP-complete. On certain classes of
graphs they can be computed in polynomial time. Such algorithms are known for
bipartite graphs, chordal graphs, cycle graphs, comparability graphs, claw-free
graphs, interval graphs and circular arc graphs for some of these problems. In
this article we introduce a new class of graphs called a layered graph and show
that if the number of vertices in a layer is then MIS,
MVC, CVC, MDS and CDC can be computed in polynomial time. The restrictions that
are employed on graph classes that admit polynomial time solutions for hard
problems, e.g. lack of cycles, bipartiteness, planarity etc. are not applicable
for this class. \\ Key words: Independent set, vertex cover, dominating set,
dynamic programming, complexity, polynomial time algorithms.Comment: 14 pages, 1 figure. A generic algorithm is given. It can be extended
to handle a wide range of hard problems. Space complexity was incorrectly
given as for MIS (identical for MVC) in the earlier version instead
of . The edges in are more clearly defined. For the
only permissible edges are where $j \in \{ i-1, i+1\}
A Linear Algorithm for Computing -set in Generalized Series-Parallel Graphs
For a graph , a set is a -set if it is a
dominating set for and each vertex is dominated by at
most two vertices of , i.e. .
Moreover a set is a total -set if for each vertex of
, it is the case that . The
-domination number of , denoted ,is the minimum
number of vertices in a -set. Every -set with cardinality of
is called a -set. Total -domination
number and -sets of are defined in a similar way. This
paper presents a linear time algorithm to find a -set and a
-set in generalized series-parallel graphs
Weighted domination number of cactus graphs
In the paper, we write a linear algorithm for calculating the weighted
domination number of a vertex-weighted cactus. The algorithm is based on the
well known depth first search (DFS) structure. Our algorithm needs less than
additions and -operations where is the number of
vertices and is the number of blocks in the cactus.Comment: 17 pages, figures, submitted to Discussiones Mathematicae Graph
Theor
Efficient Encoding of Watermark Numbers as Reducible Permutation Graphs
In a software watermarking environment, several graph theoretic watermark
methods use numbers as watermark values, where some of these methods encode the
watermark numbers as graph structures. In this paper we extended the class of
error correcting graphs by proposing an efficient and easily implemented codec
system for encoding watermark numbers as reducible permutation flow-graphs.
More precisely, we first present an efficient algorithm which encodes a
watermark number as self-inverting permutation and, then, an
algorithm which encodes the self-inverting permutation as a reducible
permutation flow-graph by exploiting domination relations on the
elements of and using an efficient DAG representation of . The
whole encoding process takes O(n) time and space, where is the binary size
of the number or, equivalently, the number of elements of the permutation
. We also propose efficient decoding algorithms which extract the number
from the reducible permutation flow-graph within the same time
and space complexity. The two main components of our proposed codec system,
i.e., the self-inverting permutation and the reducible permutation
graph , incorporate important structural properties which make our
system resilient to attacks
Parameterized Complexity of Safe Set
In this paper we study the problem of finding a small safe set in a graph
, i.e. a non-empty set of vertices such that no connected component of
is adjacent to a larger component in . We enhance our
understanding of the problem from the viewpoint of parameterized complexity by
showing that (1) the problem is W[2]-hard when parameterized by the pathwidth
and cannot be solved in time unless the ETH is false, (2) it
admits no polynomial kernel parameterized by the vertex cover number
unless , but (3) it is fixed-parameter
tractable (FPT) when parameterized by the neighborhood diversity , and (4)
it can be solved in time for some double exponential function
where is the clique-width. We also present (5) a faster FPT algorithm when
parameterized by solution size
Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1-Sperner hypergraphs
A hypergraph is said to be -Sperner if for every two hyperedges the
smallest of their two set differences is of size one. We present several
applications of -Sperner hypergraphs and their structure to graphs. In
particular, we consider the classical characterizations of threshold and
domishold graphs and use them to obtain further characterizations of these
classes in terms of -Spernerness, thresholdness, and -asummability of
their vertex cover, clique, dominating set, and closed neighborhood
hypergraphs. Furthermore, we apply a decomposition property of -Sperner
hypergraphs to derive decomposition theorems for two classes of split graphs, a
class of bipartite graphs, and a class of cobipartite graphs. These
decomposition theorems are based on certain matrix partitions of the
corresponding graphs, giving rise to new classes of graphs of bounded
clique-width and new polynomially solvable cases of several domination
problems.Comment: 31 pages, 9 figure
Algorithms for Unipolar and Generalized Split Graphs
A graph is a {\it unipolar graph} if there exits a partition such that, is a clique and induces the disjoint union of
cliques. The complement-closed class of {\it generalized split graphs} are
those graphs such that either {\it or} the complement of is
unipolar. Generalized split graphs are a large subclass of perfect graphs. In
fact, it has been shown that almost all -free (and hence, almost all
perfect graphs) are generalized split graphs. In this paper we present a
recognition algorithm for unipolar graphs that utilizes a minimal triangulation
of the given graph, and produces a partition when one exists. Our algorithm has
running time O(), where is the number of edges in a
minimal triangulation of the given graph. Generalized split graphs can
recognized via this algorithm in O(nm' + n\OL{m}') = O() time. We give
algorithms on unipolar graphs for finding a maximum independent set and a
minimum clique cover in O() time and for finding a maximum clique and a
minimum proper coloring in O(), when a unipolar partition is
given. These algorithms yield algorithms for the four optimization problems on
generalized split graphs that have the same worst-case time bound. We also
prove that the perfect code problem is NP-Complete for unipolar graphs.Comment: Please cite this article in press as: E.M. Eschen, X. Wang,
Algorithms for unipolar and generalized split graphs. Discrete Applied
Mathematics (2013),http://dx.doi.org/10.1016/j.dam.2013.0801
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