F-Sets in graphs

AbstractA subset S of the vertex set of a graph G is called an F-set if every α ϵ Γ(G), the automorphism group of G, is completely specified by specifying the images under α of all the points of S, and S has a minimum number of points. The number of points, k(G), in an F-set is an invariant of G, whose properties are studied in this paper. For a finite group Γ we define k(Γ) = max{k(G) | Γ(G) = Γ}. Graphs with a given Abelian group and given k-value (k ≤ k(Γ)) have been constructed. Graphs with a given group and k-value 1 are constructed which give simple proofs to the theorems of Frucht and Bouwer on the existence of graphs with given abstract/permutation groups

Upward-closed hereditary families in the dominance order

The majorization relation orders the degree sequences of simple graphs into posets called dominance orders. As shown by Hammer et al. and Merris, the degree sequences of threshold and split graphs form upward-closed sets within the dominance orders they belong to, i.e., any degree sequence majorizing a split or threshold sequence must itself be split or threshold, respectively. Motivated by the fact that threshold graphs and split graphs have characterizations in terms of forbidden induced subgraphs, we define a class $\mathcal{F}$ of graphs to be dominance monotone if whenever no realization of $e$ contains an element $\mathcal{F}$ as an induced subgraph, and $d$ majorizes $e$, then no realization of $d$ induces an element of $\mathcal{F}$. We present conditions necessary for a set of graphs to be dominance monotone, and we identify the dominance monotone sets of order at most 3.Comment: 15 pages, 6 figure

On the logical definability of certain graph and poset languages

We show that it is equivalent, for certain sets of finite graphs, to be definable in CMS (counting monadic second-order logic, a natural extension of monadic second-order logic), and to be recognizable in an algebraic framework induced by the notion of modular decomposition of a finite graph. More precisely, we consider the set $F\_\infty$ of composition operations on graphs which occur in the modular decomposition of finite graphs. If $F$ is a subset of $F\_{\infty}$, we say that a graph is an \calF-graph if it can be decomposed using only operations in $F$. A set of $F$-graphs is recognizable if it is a union of classes in a finite-index equivalence relation which is preserved by the operations in $F$. We show that if $F$ is finite and its elements enjoy only a limited amount of commutativity -- a property which we call weak rigidity, then recognizability is equivalent to CMS-definability. This requirement is weak enough to be satisfied whenever all $F$-graphs are posets, that is, transitive dags. In particular, our result generalizes Kuske's recent result on series-parallel poset languages

Independence densities of hypergraphs

We consider the number of independent sets in hypergraphs, which allows us to define the independence density of countable hypergraphs. Hypergraph independence densities include a broad family of densities over graphs and relational structures, such as $F$-free densities of graphs for a given graph $F.$ In the case of $k$-uniform hypergraphs, we prove that the independence density is always rational. In the case of finite but unbounded hyperedges, we show that the independence density can be any real number in $[0,1].$ Finally, we extend the notion of independence density via independence polynomials

Subspace Evasive Sets

In this work we describe an explicit, simple, construction of large subsets of F^n, where F is a finite field, that have small intersection with every k-dimensional affine subspace. Interest in the explicit construction of such sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl (2004) who showed how such constructions over the binary field can be used to construct explicit Ramsey graphs. More recently, Guruswami (2011) showed that, over large finite fields (of size polynomial in n), subspace evasive sets can be used to obtain explicit list-decodable codes with optimal rate and constant list-size. In this work we construct subspace evasive sets over large fields and use them to reduce the list size of folded Reed-Solomon codes form poly(n) to a constant.Comment: 16 page