In the graph database literature the term "join" does not refer to an operator used to merge two graphs. In particular, a counterpart of the relational join is not present in existing graph query languages, and consequently no efficient algorithms have been developed for this operator. This paper provides two main contributions. First, we define a binary graph join operator that acts on the vertices as a standard relational join and combines the edges according to a user-defined semantics. Then we propose the "CoGrouped Graph Conjunctive @math -Join" algorithm running over data indexed in secondary memory. Our implementation outperforms the execution of the same operation in Cypher and SPARQL on major existing graph database management systems by at least one order of magnitude, also including indexing and loading time.
Discrete Mathematics At the time of writing, the only field where graph joins where effectively discussed is Discrete Mathematics. In this field such operations are defined over either on finite graphs or on finite graphs with cycles, and are named @cite_2 . As the name suggests, every graph product of two graphs, e.g. @math and @math , produces a graph whose vertex set is defined as @math , while the edge set changes accordingly to the different graph product definition. Consequently the Kroneker Graph Product @cite_37 is defined as follows: while the @cite_14 is defined as follows: Please observe that this definition creates a new vertex which is a pair of vertices: hereby such operation is defined differently from the relational algebra's cartesian product, where the two vertices are merged. As a consequence, such graph products admit commutativity and associativity properties only up to graph isomorphism. Other graph products are and @cite_2 @cite_26 .
@cite_37 @cite_14 @cite_26 @cite_2
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We present an algorithm that determines the prime factors of connected graphs with respect to the Cartesian product in linear time and space. This improves a result of [Cartesian graph factorization at logarithmic cost per edge, Comput. Complexity 2 (1992) 331-349], who compute the prime factors in O(mlogn) time, where m denotes the number of vertices of G and n the number of edges. Our algorithm is conceptually simpler. It gains its efficiency by the introduction of edge-labellings. Basic Concepts. Hypercubes. Hamming Graphs. Cartesian Products. Strong and Direct Products. Lexicographic Products. Fast Recognition Algorithms. Invariants. Appendices. Bibliography. Indexes. Handbook of Product Graphs, Second Edition examines the dichotomy between the structure of products and their subgraphs. It also features the design of efficient algorithms that recognize products and their subgraphs and explores the relationship between graph parameters of the product and factors. Extensively revised and expanded, the handbook presents full proofs of many important results as well as up-to-date research and conjectures. Results and Algorithms New to the Second Edition: Cancellation results A quadratic recognition algorithm for partial cubes Results on the strong isometric dimension Computing the Wiener index via canonical isometric embedding Connectivity results A fractional version of Hedetniemis conjecture Results on the independence number of Cartesian powers of vertex-transitive graphs Verification of Vizings conjecture for chordal graphs Results on minimum cycle bases Numerous selected recent results, such as complete minors and nowhere-zero flows The second edition of this classic handbook provides a thorough introduction to the subject and an extensive survey of the field. The first three parts of the book cover graph products in detail. The authors discuss algebraic properties, such as factorization and cancellation, and explore interesting and important classes of subgraphs. The fourth part presents algorithms for the recognition of products and related classes of graphs. The final two parts focus on graph invariants and infinite, directed, and product-like graphs. Sample implementations of selected algorithms and other information are available on the books website, which can be reached via the authors home pages.