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acyclic adjacent algorithm approximation assume binary relation bipartite bipartite graph Borel space chordal graphs chromatic polynomial clique codeword cograph complexity Computer Science consider constraint construction contains convergence Corollary cycle D+DK database space decomposition defined definition deletion denote distribution invariant Dunham edge edge-graceful edge-graceful labeling edge-minimal vertex sequence elements equations example exists Figure fragment functional dependencies Gaussian elimination given graph G heuristic i=l x integer interval graphs it-combinations iteration join attributes labels latin squares Lee's frame Lemma length Let G Levenshtein distance linear mapping Math matrix method minimizing Minimum Detour node nonlinear notation NP-complete obtained optimal pair partition partitioned databases path plane procedure processes processors Proof relation schemes satisfied semijoin solution solving squares with holes steps string structure subgraph subset subspace Theorem Theory threshold graphs topological space transformation two-way join unicyclic unicyclic graph vector vertices zero