Abstract
We describe in the present work all minimal clique separators of the four standard products-Cartesian, strong, direct, and lexicographic-as well as all maximal atoms of the Cartesian, strong and lexicographic product, while we only partially describe maximal atoms of direct products. Typically, a product has no clique separator and so the product is a maximal atom.
Highlights
AND PRELIMINARIESA clique separator of a graph G with k components is a clique in G whose removal disconnects the graph into more than k components
Decomposing a graph into atoms and clique separators is a very important problem, algorithmically or otherwise, because, many hard graph problems like finding a maximum size clique can be optimized by first decomposing the graph into smaller clique separator-free graphs
We study the atoms and clique separators of all standard products
Summary
A clique separator of a graph G with k components is a clique (a subgraph consisting of pairwise adjacent vertices) in G whose removal disconnects the graph into more than k components. We wish to obtain results of that type for minimal clique separators and maximal atoms for all standard products. For this we completely describe minimal clique separating sets of all four products as well as maximal atoms of the Cartesian, the lexicographic, and the strong product, while for the direct product this task is done partially These structural characterization enable us to apply the decomposition algorithm from [4] to obtain fast decompositions in lexicographic and strong product in comparison with applying the algorithm directly. The sections are devoted to minimal clique separators and maximal atoms of the Cartesian, the lexicographic, the strong, and the direct product, respectively.
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