1,367 research outputs found

    On double domination in graphs

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    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

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    In this paper, we study a parameter that is squeezed between arguably the two important domination parameters, namely the domination number, γ(G)\gamma(G), and the total domination number, γt(G)\gamma_t(G). A set SS of vertices in GG is a semitotal dominating set of GG if it is a dominating set of GG and every vertex in S is within distance 22 of another vertex of SS. The semitotal domination number, γt2(G)\gamma_{t2}(G), is the minimum cardinality of a semitotal dominating set of GG. We observe that γ(G)γt2(G)γt(G)\gamma(G)\leq \gamma_{t2}(G)\leq \gamma_t(G). 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

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    summary:Let GG be a 33-connected triangulated disc of order nn with the boundary cycle CC of the outer face of GG. Tokunaga (2013) conjectured that GG has a dominating set of cardinality at most 14(n+2)\frac 14(n+2). This conjecture is proved in Tokunaga (2020) for GCG-C being a tree. In this paper we prove the above conjecture for GCG-C 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

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    A set SS of vertices in a graph G=(V,E)G = (V,E) is a 22-dominating set if every vertex of VSV\setminus S is adjacent to at least two vertices of SS. The 22-domination number of a graph GG, denoted by γ2(G)\gamma_2(G), is the minimum size of a 22-dominating set of GG. The 22-domination subdivision number sdγ2(G)sd_{\gamma_2}(G) is the minimum number of edges that must be subdivided (each edge in GG can be subdivided at most once) in order to increase the 22-domination number. The authors have recently proved that for any tree TT of order at least 33, 1sdγ2(T)21 \leq sd_{\gamma_2}(T)\leq 2. In this paper we provide a constructive characterization of the trees whose 22-domination subdivision number is 22

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    Domination parameters with number 2: Interrelations and algorithmic consequences

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    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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