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 $2$-domination number, $\gamma_{w2}(G)$, the $2$-domination number, $\gamma_2(G)$, the $\{2\}$-domination number, $\gamma_{\{2\}}(G)$, the double domination number, $\gamma_{\times 2}(G)$, the total $\{2\}$-domination number, $\gamma_{t\{2\}}(G)$, and the total double domination number, $\gamma_{t\times 2}(G)$, where $G$ 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, $\gamma_R(G)$, and two classical parameters, the domination number, $\gamma(G)$, and the total domination number, $\gamma_t(G)$, we consider 13 domination invariants in graphs $G$. 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

A New Optimality Measure for Distance Dominating Sets

  We study the problem of finding the smallest power of an input graph that has k disjoint dominating sets, where the ith power of an input graph G is constructed by adding edges between pairs of vertices in G at distance i or less, and a subset of vertices in a graph G is a dominating set if and only if every vertex in G is adjacent to a vertex in this subset.   The problem is a different view of the d-domatic number problem in which the goal is to find the maximum number of disjoint dominating sets in the dth power of the input graph.   This problem is motivated by applications in multi-facility location and distributed networks. In the facility location framework, for instance, there are k types of services that all clients in different regions of a city should receive. A graph representing the map of regions in the city is given where the nodes of the graph represent regions and neighboring regions are connected by edges. The problem is how to establish facility servers in the city (each region can host at most one server) such that every client in the city can access a facility server in its region or in a region in the neighborhood. Since it may not be possible to find a facility location satisfying this condition, "a region in the neighborhood" required in the question is modified to "a region at the minimum possible distance d".   In this thesis, we study the connection of the above-mentioned problem with similar problems including the domatic number problem and the d-domatic number problem. We show that the problem is NP-complete for any fixed k greater than two even when the input graph is restricted to split graphs, 2-connected graphs, or planar bipartite graphs of degree four. In addition, the problem is in P for bounded tree-width graphs, when considering k as a constant, and for strongly chordal graphs, for any k. Then, we provide a slightly simpler proof for a known upper bound for the problem. We also develop an exact (exponential) algorithm for the problem, running in time O(2. 73n). Moreover, we prove that the problem cannot be approximated within ratio smaller than 2 even for split graphs, 2-connected graphs, and planar bipartite graphs of degree four. We propose a greedy 3-approximation algorithm for the problem in the general case, and other approximation ratios for permutation graphs, distance-hereditary graphs, cocomparability graphs, dually chordal graphs, and chordal graphs. Finally, we list some directions for future work