1,367 research outputs found
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
Semitotal domination in trees
In this paper, we study a parameter that is squeezed between arguably the two
important domination parameters, namely the domination number, , and
the total domination number, . A set of vertices in is a
semitotal dominating set of if it is a dominating set of and every
vertex in S is within distance of another vertex of . The semitotal
domination number, , is the minimum cardinality of a semitotal
dominating set of . We observe that . In this paper, we give a lower bound for the semitotal domination
number of trees and we characterize the extremal trees. In addition, we
characterize trees with equal domination and semitotal domination numbers.Comment: revise
On the domination of triangulated discs
summary:Let be a -connected triangulated disc of order with the boundary cycle of the outer face of . Tokunaga (2013) conjectured that has a dominating set of cardinality at most . This conjecture is proved in Tokunaga (2020) for being a tree. In this paper we prove the above conjecture for being a unicyclic graph. We also deduce some bounds for the double domination number, total domination number and double total domination number in triangulated discs
Trees whose 2-domination subdivision number is 2
A set of vertices in a graph is a -dominating set if every vertex of is adjacent to at least two vertices of . The -domination number of a graph , denoted by , is the minimum size of a -dominating set of . The -domination subdivision number is the minimum number of edges that must be subdivided (each edge in can be subdivided at most once) in order to increase the -domination number. The authors have recently proved that for any tree of order at least , . In this paper we provide a constructive characterization of the trees whose -domination subdivision number is
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
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