895 research outputs found
Efficient algorithms for tuple domination on co-biconvex graphs and web graphs
A vertex in a graph dominates itself and each of its adjacent vertices. The
-tuple domination problem, for a fixed positive integer , is to find a
minimum sized vertex subset in a given graph such that every vertex is
dominated by at least k vertices of this set. From the computational point of
view, this problem is NP-hard. For a general circular-arc graph and ,
efficient algorithms are known to solve it (Hsu et al., 1991 & Chang, 1998) but
its complexity remains open for . A -matrix has the consecutive
0's (circular 1's) property for columns if there is a permutation of its rows
that places the 0's (1's) consecutively (circularly) in every column.
Co-biconvex (concave-round) graphs are exactly those graphs whose augmented
adjacency matrix has the consecutive 0's (circular 1's) property for columns.
Due to A. Tucker (1971), concave-round graphs are circular-arc. In this work,
we develop a study of the -tuple domination problem on co-biconvex graphs
and on web graphs which are not comparable and, in particular, all of them
concave-round graphs. On the one side, we present an -time algorithm
for solving it for each , where is the set of universal
vertices and the total number of vertices of the input co-biconvex graph.
On the other side, the study of this problem on web graphs was already started
by Argiroffo et al. (2010) and solved from a polyhedral point of view only for
the cases and , where equals the degree of each vertex of
the input web graph . We complete this study for web graphs from an
algorithmic point of view, by designing a linear time algorithm based on the
modular arithmetic for integer numbers. The algorithms presented in this work
are independent but both exploit the circular properties of the augmented
adjacency matrices of each studied graph class.Comment: 21 pages, 7 figures. Keywords: -tuple dominating sets, augmented
adjacency matrices, stable sets, modular arithmeti
Domination parameters with number 2: Interrelations and algorithmic consequences
In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak 2-domination number, γw2(G), the 2-domination number, γ2(G), the {2}-domination number, γ{2}(G), the double domination number, γ×2(G), the total {2}-domination number, γt{2}(G), and the total double domination number, γt×2(G), where G is a graph in which the corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, γR(G), and two classical parameters, the domination number, γ(G), and the total domination number, γt(G), we consider 13 domination invariants in graphs. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, a large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain new complexity results regarding the existence of approximation algorithms for the studied invariants, matched with tight or almost tight inapproximability bounds, which hold even in the class of split graphs.Fil: Bonomo, Flavia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Brešar, Boštjan. Institute of Mathematics, Physics and Mechanics; Eslovenia. University of Maribor; EsloveniaFil: Grippo, Luciano Norberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Milanič, Martin. University of Primorska; EsloveniaFil: Safe, Martin Dario. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentin
Domination parameters with number 2: interrelations and algorithmic consequences
In this paper, we study the most basic domination invariants in graphs, in
which number 2 is intrinsic part of their definitions. We classify them upon
three criteria, two of which give the following previously studied invariants:
the weak -domination number, , the -domination number,
, the -domination number, , the double
domination number, , the total -domination number,
, and the total double domination number, , where is a graph in which a corresponding invariant is well
defined. The third criterion yields rainbow versions of the mentioned six
parameters, one of which has already been well studied, and three other give
new interesting parameters. Together with a special, extensively studied Roman
domination, , and two classical parameters, the domination number,
, and the total domination number, , we consider 13
domination invariants in graphs . In the main result of the paper we present
sharp upper and lower bounds of each of the invariants in terms of every other
invariant, large majority of which are new results proven in this paper. As a
consequence of the main theorem we obtain some complexity results for the
studied invariants, in particular regarding the existence of approximation
algorithms and inapproximability bounds.Comment: 45 pages, 4 tables, 7 figure
(Total) Vector Domination for Graphs with Bounded Branchwidth
Given a graph of order and an -dimensional non-negative
vector , called demand vector, the vector domination
(resp., total vector domination) is the problem of finding a minimum
such that every vertex in (resp., in ) has
at least neighbors in . The (total) vector domination is a
generalization of many dominating set type problems, e.g., the dominating set
problem, the -tuple dominating set problem (this is different from the
solution size), and so on, and its approximability and inapproximability have
been studied under this general framework. In this paper, we show that a
(total) vector domination of graphs with bounded branchwidth can be solved in
polynomial time. This implies that the problem is polynomially solvable also
for graphs with bounded treewidth. Consequently, the (total) vector domination
problem for a planar graph is subexponential fixed-parameter tractable with
respectto , where is the size of solution.Comment: 16 page
On the approximability and exact algorithms for vector domination and related problems in graphs
We consider two graph optimization problems called vector domination and
total vector domination. In vector domination one seeks a small subset S of
vertices of a graph such that any vertex outside S has a prescribed number of
neighbors in S. In total vector domination, the requirement is extended to all
vertices of the graph. We prove that these problems (and several variants
thereof) cannot be approximated to within a factor of clnn, where c is a
suitable constant and n is the number of the vertices, unless P = NP. We also
show that two natural greedy strategies have approximation factors ln D+O(1),
where D is the maximum degree of the input graph. We also provide exact
polynomial time algorithms for several classes of graphs. Our results extend,
improve, and unify several results previously known in the literature.Comment: In the version published in DAM, weaker lower bounds for vector
domination and total vector domination were stated. Being these problems
generalization of domination and total domination, the lower bounds of 0.2267
ln n and (1-epsilon) ln n clearly hold for both problems, unless P = NP or NP
\subseteq DTIME(n^{O(log log n)}), respectively. The claims are corrected in
the present versio
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