4,951 research outputs found
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
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
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
Rainbow Connection Number and Connected Dominating Sets
Rainbow connection number rc(G) of a connected graph G is the minimum number
of colours needed to colour the edges of G, so that every pair of vertices is
connected by at least one path in which no two edges are coloured the same. In
this paper we show that for every connected graph G, with minimum degree at
least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2,
where {\gamma}_c(G) is the connected domination number of G. Bounds of the form
diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special
graph classes follow as easy corollaries from this result. This includes
interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and
chain graphs all with minimum degree at least 2 and connected. We also show
that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of
these cases, we also demonstrate the tightness of the bounds. An extension of
this idea to two-step dominating sets is used to show that for every connected
graph on n vertices with minimum degree {\delta}, the rainbow connection number
is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of
Schiermeyer (2009), improving the previously best known bound of 20n/{\delta}
by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up
to additive factors by a construction of Caro et al. (2008).Comment: 14 page
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
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