981 research outputs found
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
Grouped Domination Parameterized by Vertex Cover, Twin Cover, and Beyond
A dominating set of graph is called an -grouped dominating set if
can be partitioned into such that the size of each
unit is and the subgraph of induced by is connected. The
concept of -grouped dominating sets generalizes several well-studied
variants of dominating sets with requirements for connected component sizes,
such as the ordinary dominating sets (), paired dominating sets (),
and connected dominating sets ( is arbitrary and ). In this paper, we
investigate the computational complexity of -Grouped Dominating Set, which
is the problem of deciding whether a given graph has an -grouped dominating
set with at most units. For general , the problem is hard to solve in
various senses because the hardness of the connected dominating set is
inherited. We thus focus on the case in which is a constant or a parameter,
but we see that the problem for every fixed is still hard to solve. From
the hardness, we consider the parameterized complexity concerning well-studied
graph structural parameters. We first see that it is fixed-parameter tractable
for and treewidth, because the condition of -grouped domination for a
constant can be represented as monadic second-order logic (mso2). This is
good news, but the running time is not practical. We then design an
-time algorithm for general
, where is the twin cover number, which is a parameter between
vertex cover number and clique-width. For paired dominating set and trio
dominating set, i.e., , we can speed up the algorithm, whose
running time becomes . We further argue the relationship
between FPT results and graph parameters, which draws the parameterized
complexity landscape of -Grouped Dominating Set.Comment: 23 pages, 6 figure
Upper paired domination versus upper domination
A paired dominating set is a dominating set with the additional property
that has a perfect matching. While the maximum cardainality of a minimal
dominating set in a graph is called the upper domination number of ,
denoted by , the maximum cardinality of a minimal paired dominating
set in is called the upper paired domination number of , denoted by
. By Henning and Pradhan (2019), we know that
for any graph without isolated vertices. We
focus on the graphs satisfying the equality . We
give characterizations for two special graph classes: bipartite and unicyclic
graphs with by using the results of Ulatowski
(2015). Besides, we study the graphs with and a
restricted girth. In this context, we provide two characterizations: one for
graphs with and girth at least 6 and the other for
-free cactus graphs with . We also pose the
characterization of the general case of -free graphs with as an open question
On double domination in graphs
In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ ×2(G). A function f(p) is defined, and it is shown that γ ×2(G) = minf(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1,…,pn) | pi ∈ IR, 0 ≤ pi ≤ 1,i = 1,…,n}. Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then γ×2(G) ≤ ((ln(1+d)+lnδ+1)/δ)n
Locating-dominating sets and identifying codes in graphs of girth at least 5
Locating-dominating sets and identifying codes are two closely related
notions in the area of separating systems. Roughly speaking, they consist in a
dominating set of a graph such that every vertex is uniquely identified by its
neighbourhood within the dominating set. In this paper, we study the size of a
smallest locating-dominating set or identifying code for graphs of girth at
least 5 and of given minimum degree. We use the technique of vertex-disjoint
paths to provide upper bounds on the minimum size of such sets, and construct
graphs who come close to meet these bounds.Comment: 20 pages, 9 figure
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