Computing the Atom Graph of a Graph and the Union Join Graph of a Hypergraph

The atom graph of a graph is a graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all possible atom trees of this graph. We provide two efficient algorithms for computing this atom graph, with a complexity in O(min(nωlogn,nm,n(n+m¯)) time, where n is the number of vertices of G, m is the number of its edges, m¯ is the number of edges of the complement of G, and ω, also denoted by α in the literature, is a real number, such that O(nω) is the best known time complexity for matrix multiplication, whose current value is 2,3728596. This time complexity is no more than the time complexity of computing the atoms in the general case. We extend our results to α-acyclic hypergraphs, which are hypergraphs having at least one join tree, a join tree of an hypergraph being defined by its hyperedges in the same way as an atom tree of a graph is defined by its atoms. We introduce the notion of union join graph, which is the union of all possible join trees; we apply our algorithms for atom graphs to efficiently compute union join graphs.

Covering Minimal Separators and Potential Maximal Cliques in $P_t$-Free Graphs

A graph is called $P_t$-free if it does not contain a $t$-vertex path as an induced subgraph. While $P_4$-free graphs are exactly cographs, the structure of $P_t$-free graphs for $t \geqslant 5$ remains little understood. On one hand, classic computational problems such as Maximum Weight Independent Set (MWIS) and $3$-Coloring are not known to be NP-hard on $P_t$-free graphs for any fixed $t$. On the other hand, despite significant effort, polynomial-time algorithms for MWIS in $P_6$-free graphs~[SODA 2019] and $3$-Coloring in $P_7$-free graphs~[Combinatorica 2018] have been found only recently. In both cases, the algorithms rely on deep structural insights into the considered graph classes. One of the main tools in the algorithms for MWIS in $P_5$-free graphs~[SODA 2014] and in $P_6$-free graphs~[SODA 2019] is the so-called Separator Covering Lemma that asserts that every minimal separator in the graph can be covered by the union of neighborhoods of a constant number of vertices. In this note we show that such a statement generalizes to $P_7$-free graphs and is false in $P_8$-free graphs. We also discuss analogues of such a statement for covering potential maximal cliques with unions of neighborhoods.

AMP Chain Graphs: Minimal Separators and Structure Learning Algorithms

This paper deals with chain graphs (CGs) under the Andersson–Madigan–Perlman (AMP) interpretation. We address the problem of finding a minimal separator in an AMP CG, namely, finding a set Z of nodes that separates a given non-adjacent pair of nodes such that no proper subset of Z separates that pair. We analyze several versions of this problem and offer polynomial time algorithms for each. These include finding a minimal separator from a restricted set of nodes, finding a minimal separator for two given disjoint sets, and testing whether a given separator is minimal. To address the problem of learning the structure of AMP CGs from data, we show that the PC-like algorithm is order dependent, in the sense that the output can depend on the order in which the variables are given. We propose several modifications of the PC-like algorithm that remove part or all of this order-dependence. We also extend the decomposition-based approach for learning Bayesian networks (BNs) to learn AMP CGs, which include BNs as a special case, under the faithfulness assumption. We prove the correctness of our extension using the minimal separator results. Using standard benchmarks and synthetically generated models and data in our experiments demonstrate the competitive performance of our decomposition-based method, called LCD-AMP, in comparison with the (modified versions of) PC-like algorithm. The LCD-AMP algorithm usually outperforms the PC-like algorithm, and our modifications of the PC-like algorithm learn structures that are more similar to the underlying ground truth graphs than the original PC-like algorithm, especially in high-dimensional settings. In particular, we empirically show that the results of both algorithms are more accurate and stabler when the sample size is reasonably large and the underlying graph is sparse

Research on Selecting a Sublattice Based on L-Context

In this paper, given a binary relation, we represented the relationship between a fuzzy graph and a fuzzy concept lattice. We introduced one of the most useful notions in Graph Theory--minimal separator. In order to make decision-making much easier, the number of concepts can be reduced by selecting a sublattice via saturating the minimal separator of a given concept, a method also proposed when converting L-context to classical context. In the end, we discussed a few open issues.

Minimal Separators

A separator of a connected graph G is a set of vertices whose removal disconnects G. In this paper we give various conditions for a separator to contain a minimal one. In particular we prove that every separator of a connected graph that has no thick end, or which is of bounded degree, contains a minimal separator.