31 research outputs found
Geodesic packing in graphs
Given a graph , a geodesic packing in is a set of vertex-disjoint
maximal geodesics, and the geodesic packing number of , {\gpack}(G), is
the maximum cardinality of a geodesic packing in . It is proved that the
decision version of the geodesic packing number is NP-complete. We also
consider the geodesic transversal number, , which is the minimum
cardinality of a set of vertices that hit all maximal geodesics in . While
\gt(G)\ge \gpack(G) in every graph , the quotient is investigated. By using the rook's graph, it is proved that there
does not exist a constant such that would hold for all graphs . If is a tree, then it is
proved that , and a linear algorithm for
determining is derived. The geodesic packing number is also
determined for the strong product of paths
Grundy domination and zero forcing in Kneser graphs
In this paper, we continue the investigation of different types of (Grundy) dominating sequences. We consider four different types of Grundy domination numbers and the related zero forcing numbers, focusing on these numbers in the well-known class of Kneser graphs Kn,r. In particular, we establish that the Grundy total domination number γ t gr(Kn,r) equals 2r r for any r ≥ 2 and n ≥ 2r + 1. For the Grundy domination number of Kneser graphs we get γgr(Kn,r) = α(Kn,r) whenever n is sufficiently larger than r. On the other hand, the zero forcing number Z(Kn,r) is proved to be n r − 2r r when n ≥ 3r + 1 and r ≥ 2, while lower and upper bounds are provided for Z(Kn,r) when 2r + 1 ≤ n ≤ 3r. Some lower bounds for different types of minimum ranks of Kneser graphs are also obtained along the way.Fil: Bresar, Bostjan. University of Maribor; Eslovenia. Institute Of Mathematics, Physics And Mechanics Ljubljana; EsloveniaFil: Kos, Tim. Institute Of Mathematics, Physics And Mechanics Ljubljana; EsloveniaFil: Torres, Pablo Daniel. Universidad Nacional de Rosario. Facultad de Ciencias Exactas IngenierÃa y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Rosario; 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
Grundy dominating sequences and zero forcing sets
In a graph a sequence of vertices is Grundy
dominating if for all we have and is Grundy total dominating if for all
we have .
The length of the longest Grundy (total) dominating sequence has
been studied by several authors. In this paper we introduce two
similar concepts when the requirement on the neighborhoods is
changed to or
. In the former case we
establish a strong connection to the zero forcing number of a graph,
while we determine the complexity of the decision problem in the
latter case. We also study the relationships among the four
concepts, and discuss their computational complexities
On Vizing's conjecture
A dominating set D for a graph G is a subset of V(G) such that any vertex in V(G)-D has a neighbor in D, and a domination number γ(G) is the size of a minimum dominating set for G. For the Cartesian product G ⃞ H Vizing's conjecture [10] states that γ(G ⃞ H) ≥ γ(G)γ(H) for every pair of graphs G,H. In this paper we introduce a new concept which extends the ordinary domination of graphs, and prove that the conjecture holds when γ(G) = γ(H) = 3