On efficient total domination

An efficiently total dominating set of a graph G is a subset of its vertices such that each vertex of G is adjacent to exactly one vertex of the subset. If there is such a subset, then G is an efficiently total dominatable graph (G is etd). We show that the corresponding etd decision problem is NP-complete on (1,2)-colorable chordal graphs and on planar bipartite graphs of maximum degree 3 and obtain polynomial solvability on T_3-free chordal graphs, implying polynomial solvability on interval graphs and circular arc graphs

Efficient total domination in digraphs

We generalize the concept of efficient total domination from graphs to digraphs. An efficiently total dominating set X of a digraph D is a vertex subset such that every vertex of D has exactly one predecessor in X . Not every digraph has an efficiently total dominating set. We study graphs that permit an orientation having such a set and give complexity results and characterizations concerning this question. Furthermore, we study the computational complexity of the (weighted) efficient total domination problem for several digraph classes. In particular we deal with most of the common generalizations of tournaments, like locally semicomplete and arc-locally semicomplete digraphs

Efficient domination in knights graphs

The influence of a vertex set S ⊆V(G) is I(S) = Σv∈S(1 + deg(v)) = Σv∈S |N[v]|, which is the total amount of domination done by the vertices in S. The efficient domination number F(G) of a graph G is equal to the maximum influence of a packing, that is, F(G) is the maximum number of vertices one can dominate under the restriction that no vertex gets dominated more than once. In this paper, we consider the efficient domination number of some finite and infinite knights chessboard graphs

A Linear Kernel for Planar Total Dominating Set

A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set.Comment: 33 pages, 13 figure

Generating cosmological perturbations with mass variations

We study the possibility that large scale cosmological perturbations have been generated during the domination and decay of a massive particle species whose mass depends on the expectation value of a light scalar field. We discuss the constraints that must be imposed on the field in order to remain light and on the annihilation cross section and decay rate of the massive particles in order for the mechanism to be efficient. We compute the resulting curvature perturbations after the mass domination, recovering the results of Dvali, Gruzinov, and Zaldarriaga in the limit of total domination. By comparing the amplitude of perturbations generated by the mass domination to those originally present from inflation, we conclude that this mechanism can be the primary source of perturbations only if inflation does not rely on slow-roll conditions.Comment: 8 pages. Proceeding of the workshop: `The Density Perturbation in the Universe', Demokritos Center, Athens, Grece, June 200

Partitioning the vertex set of GG to make G □ HG\,\Box\, H an efficient open domination graph

A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs $G$ for which the Cartesian product $G \Box H$ is an efficient open domination graph when $H$ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of $V(G)$. For the class of trees when $H$ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products $G \Box H$ when $H$ is a 5-cycle or a 4-cycle.Comment: 16 pages, 2 figure