A notion of degeneration of elements in groups is introduced. It is used to parametrize the orbits in a finite abelian group under its full automorphism group by a finite distributive lattice. A pictorial description of this lattice leads to an intuitive self-contained exposition of some of the basic facts concerning these orbits, including their enumeration. Given a partition $\lambda$, the lattice parametrizing orbits in a finite abelian p-group of type $\lambda$ is found to be independent of p. The order of the orbit corresponding to each parameter, which turns out to be a polynomial in p, is calculated. The description of orbits is extended to subquotients by certain characteristic subgroups. Each such characteristic subquotient is shown to have a unique maximal orbit.Comment: 14 pages, 5 figure
a b s t r a c tA subset D ⊆ V of a graph G = (V , E) is a (1, j)-set (Chellali et al., 2013) if every vertex v ∈ V \D is adjacent to at least 1 but not more than j vertices in D. The cardinality of a minimum (1, j)-set of G, denoted as γ (1,j) (G), is called the (1, j)-domination number of G. In this paper, using probabilistic methods, we obtain an upper bound on γ (1,j) (G) for j ≥ O(log ∆), where ∆ is the maximum degree of the graph. The proof of this upper bound yields a randomized linear time algorithm. We show that the associated decision problem is NP-complete for choral graphs but, answering a question of Chellali et al., provide a linear-time algorithm for trees for a fixed j. Apart from this, we design a polynomial time algorithm for finding γ (1,j) (G) for a fixed j in a split graph, and show that (1, j)-set problem is fixed parameter tractable in bounded genus graphs and bounded treewidth graphs.
Showing the existence of small-sized-nets has been the subject of investigation for almost 30 years, starting from the breakthrough of Haussler and Welzl (1987). Following a long line of successive improvements, recent results have settled the question of the size of the smallest-nets for set systems as a function of their so-called shallow cell complexity. In this paper we give a short proof of this theorem in the space of a few elementary paragraphs. This immediately implies all known cases of results on unweighted-nets studied for the past 28 years, starting from the result of Matousek, Seidel and Welzl (SoCG 1990) to that of Clarkson and Varadajan (DCG 2007) to that of Varadarajan (STOC 2010) and Chan et al. (SODA 2012) for the unweighted case, as well as the technical and intricate paper of Aronov et al. (SIAM Journal on Computing, 2010). We find it quite surprising that such a simple elementary approach was missed in all earlier work, as all ingredients were already known in 1991.
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