k-Efficient Partitions of Graphs

A set S = {u1, u2,..., ut} of vertices of G is an efficient dominating set if every vertex of G is dominated exactly once by the vertices of S. Letting Ui denote the set of vertices dominated by ui, we note that {U1, U2,... Ut} is a partition of the vertex set of G and that each Ui contains the vertex ui and all the vertices at distance 1 from it in G. In this paper, we generalize the concept of efficient domination by considering k-efficient domination partitions of the vertex set of G, where each element of the partition is a set consisting of a vertex ui and all the vertices at distance di from it, where di ∈ {0, 1,..., k}. For any integer k ≥ 0, the k-efficient domination number of G equals the minimum order of a k-efficient partition of G. We determine bounds on the k-efficient domination number for general graphs, and for k ∈ {1, 2}, we give exact values for some graph families. Complexity results are also obtained

An Efficient Algorithm for Power Dominating Set

The problem Power Dominating Set (PDS) is motivated by the placement of phasor measurement units to monitor electrical networks. It asks for a minimum set of vertices in a graph that observes all remaining vertices by exhaustively applying two observation rules. Our contribution is twofold. First, we determine the parameterized complexity of PDS by proving it is W[P]-complete when parameterized with respect to the solution size. We note that it was only known to be W[2]-hard before. Our second and main contribution is a new algorithm for PDS that efficiently solves practical instances. Our algorithm consists of two complementary parts. The first is a set of reduction rules for PDS that can also be used in conjunction with previously existing algorithms. The second is an algorithm for solving the remaining kernel based on the implicit hitting set approach. Our evaluation on a set of power grid instances from the literature shows that our solver outperforms previous state-of-the-art solvers for PDS by more than one order of magnitude on average. Furthermore, our algorithm can solve previously unsolved instances of continental scale within a few minutes

Playing with parameters: structural parameterization in graphs

When considering a graph problem from a parameterized point of view, the parameter chosen is often the size of an optimal solution of this problem (the "standard" parameter). A natural subject for investigation is what happens when we parameterize such a problem by various other parameters, some of which may be the values of optimal solutions to different problems. Such research is known as parameterized ecology. In this paper, we investigate seven natural vertex problems, along with their respective parameters: the size of a maximum independent set, the size of a minimum vertex cover, the size of a maximum clique, the chromatic number, the size of a minimum dominating set, the size of a minimum independent dominating set and the size of a minimum feedback vertex set. We study the parameterized complexity of each of these problems with respect to the standard parameter of the others.Comment: 17 page

Fast algorithms for min independent dominating set

We first devise a branching algorithm that computes a minimum independent dominating set on any graph with running time O*(2^0.424n) and polynomial space. This improves the O*(2^0.441n) result by (S. Gaspers and M. Liedloff, A branch-and-reduce algorithm for finding a minimum independent dominating set in graphs, Proc. WG'06). We then show that, for every r>3, it is possible to compute an r-((r-1)/r)log_2(r)-approximate solution for min independent dominating set within time O*(2^(nlog_2(r)/r))