Krausz dimension and its generalizations in special graph classes

A {\it krausz $(k,m)$-partition} of a graph $G$ is the partition of $G$ into cliques, such that any vertex belongs to at most $k$ cliques and any two cliques have at most $m$ vertices in common. The {\it $m$-krausz} dimension $kdim_m(G)$ of the graph $G$ is the minimum number $k$ such that $G$ has a krausz $(k,m)$-partition. 1-krausz dimension is known and studied krausz dimension of graph $kdim(G)$. In this paper we prove, that the problem $"kdim(G)\leq 3"$ is polynomially solvable for chordal graphs, thus partially solving the problem of P. Hlineny and J. Kratochvil. We show, that the problem of finding $m$-krausz dimension is NP-hard for every $m\geq 1$, even if restricted to (1,2)-colorable graphs, but the problem $"kdim_m(G)\leq k"$ is polynomially solvable for $(\infty,1)$-polar graphs for every fixed $k,m\geq 1$

Bell monogamy relations in arbitrary qubit networks

Characterizing trade-offs between simultaneous violations of multiple Bell inequalities in a large network of qubits is computationally demanding. We propose a graph-theoretic approach to efficiently produce Bell monogamy relations in arbitrary arrangements of qubits. All the relations obtained for bipartite Bell inequalities are tight and leverage only a single Bell monogamy relation. This feature is unique to bipartite Bell inequalities, as we show that there is no finite set of such elementary monogamy relations for multipartite inequalities. Nevertheless, many tight monogamy relations for multipartite inequalities can be obtained with our method as shown in explicit examples

Bus interconnection networks

AbstractIn bus interconnection networks every bus provides a communication medium between a set of processors. These networks are modeled by hypergraphs where vertices represent the processors and edges represent the buses. We survey the results obtained on the construction methods that connect a large number of processors in a bus network with given maximum processor degree Δ, maximum bus size r, and network diameter D. (In hypergraph terminology this problem is known as the (Δ,D, r)-hypergraph problem.)The problem for point-to-point networks (the case r = 2) has been extensively studied in the literature. As a result, several families of networks have been proposed. Some of these point-to-point networks can be used in the construction of bus networks. One approach is to consider the dual of the network. We survey some families of bus networks obtained in this manner. Another approach is to view the point-to-point networks as a special case of the bus networks and to generalize the known constructions to bus networks. We provide a summary of the tools developed in the theory of hypergraphs and directed hypergraphs to handle this approach

HYTREL: Hypergraph-enhanced Tabular Data Representation Learning

Language models pretrained on large collections of tabular data have demonstrated their effectiveness in several downstream tasks. However, many of these models do not take into account the row/column permutation invariances, hierarchical structure, etc. that exist in tabular data. To alleviate these limitations, we propose HYTREL, a tabular language model, that captures the permutation invariances and three more structural properties of tabular data by using hypergraphs - where the table cells make up the nodes and the cells occurring jointly together in each row, column, and the entire table are used to form three different types of hyperedges. We show that HYTREL is maximally invariant under certain conditions for tabular data, i.e., two tables obtain the same representations via HYTREL iff the two tables are identical up to permutations. Our empirical results demonstrate that HYTREL consistently outperforms other competitive baselines on four downstream tasks with minimal pretraining, illustrating the advantages of incorporating the inductive biases associated with tabular data into the representations. Finally, our qualitative analyses showcase that HYTREL can assimilate the table structures to generate robust representations for the cells, rows, columns, and the entire table.Comment: NeurIPS 2023 (spotlight

Характеризация и распознавание графов пересечений ребер трихроматических гиперграфов ограниченной кратности в классе расщепляемых графов

A hypergraph is called k-chromatic if its vertex set can be partitioned into at most k pairwise disjoint subsets when each subset has no more than two common vertices with every edge of the hypergraph. The multiplicity of a pair of vertices in a hypergraph is the number of hypergraph edges containing the pair of vertices. The multiplicity of a hypergraph is the maximum multiplicity of the pairs of vertices. Let Lm(k) denote the class of edge intersection graphs of k-chromatic hypergraphs with multiplicity at most m. It is known that the problem of recognizing graphs from L1(k) is polynomially solvable if k = 2 and is NP-complete if k = 3. The complexity of the recognition of graphs from Lm(k) for fixed k ≥ 2 and m ≥ 2 is currently unknown.A split graph is a graph whose vertices can be partitioned into a clique and an independent set. It is known that for any k ≥ 2 the graphs from L1(k) can be characterized by a finite list of forbidden induced subgraphs in the class of split graphs. It was earlier proved that there exists a finite characterization in terms of forbidden induced subgraphs for the graphs from L2(3) in the class of split graphs.It is proved in the article that a finite characterization in terms of forbidden induced subgraphs for the graphs from Lm(3) (for fixed m ≥ 2) exists in the class of split graphs. In particular, it follows that the problem of recognizing graphs from Lm(3), m ≥ 2 is polynomially solvable in the class of split graphs.Гиперграф, множество вершин которого можно разбить не более чем на к попарно непересекающихся подмножеств, каждое из которых имеет не более одной общей вершины с любым ребром гиперграфа, называется к-хроматическим. Кратность пары вершин гиперграфа - это число его ребер, содержащих обе вершины пары, тогда кратность гиперграфа - это максимальная кратность пар его вершин. Пусть Lm(k) обозначает класс графов пересечений ребер к-хроматических гиперграфов кратности не выше m. Задача распознавания графов из L1(k) полиномиально разрешима при к = 2 и является NP-полной при к = 3. Вопрос о сложности распознавания графов из Lm(k) при фиксированных к ≥ 2 и m ≥ 2 в настоящее время остается открытым.Граф называется расщепляемым, если множество его вершин можно разбить на два непересекающихся подмножества, одно из которых является кликой, а другое - независимым множеством вершин. Для любого к ≥ 2 графы из L1(k) характеризуются конечным списком запрещенных порожденных подграфов в классе расщепляемых графов.Ранее было доказано [1], что для графов из L2(3) существует конечная характеризация в терминах запрещенных порожденных подграфов в классе расщепляемых графов. В настоящей работе этот результат обобщен на класс Lm(3), где фиксированное m ≥ 2. Важным следствием основного результата работы является полиномиальная разрешимость задачи распознавания Gϵ Lm(3) в классе расщепляемых графов.

Line Directed Hypergraphs

In this article we generalize the concept of line digraphs to line dihypergraphs. We give some general properties in particular concerning connectivity parameters of dihypergraphs and their line dihypergraphs, like the fact that the arc connectivity of a line dihypergraph is greater than or equal to that of the original dihypergraph. Then we show that the De Bruijn and Kautz dihypergraphs (which are among the best known bus networks) are iterated line digraphs. Finally we give short proofs that they are highly connected

Line hypergraphs

AbstractIn this paper, we introduce a new multivalued function L called the line hypergraph. The function L generalizes two classical concepts at once, namely, of the line graph and the dual hypergraph. In terms of this function, proofs of some known theorems on line graphs can be unified and their more general versions can be obtained. Three such theorems are considered here, namely, the Berge theorem describing all hypergraphs with a given line graph G in terms of clique coverings of G (Berge, 1973, p. 400), the Krausz global characterization of line graphs for simple graphs (Krausz, 1943) and the Whitney theorem on isomorphisms of line graphs (Whitney, 1932)