Abstract
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.
Highlights
We focus on the atom graph, whose vertices are the atoms and whose edges are those of all possible atom trees
We introduce the notion of union join graph, which is the union of all join trees, and provide algorithms to compute this object efficiently
Given an atom tree T of G and its subset relation sub, it scans the edges of T, and for each edge AB, it computes the set of edges of the atom graph associated with the minimal separator S associated with AB if it has not been computed yet, i.e., if AB does not belong to the set of edges computed so far
Summary
Our first goal in this paper is to propose efficient algorithms to compute the atom graph, both in the general case and in the case of chordal graphs. One takes as input an atom tree as well as the inclusion relation between the separators represented by its edges, and the other takes as input the weighted intersection graph of the atoms In both cases, we provide an O(n2 ) algorithm to compute the atom graph from the input. We introduce the notion of union join graph, which is the union of all join trees, and provide algorithms to compute this object efficiently.
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