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1-factors 2-essential 2-packing 3-critical 5-halving adjacent assume bicovering bipartite graph blocks cardinality Cartesian product Cayley graph cell color columns Combinatorial complete 4-partite graphs complete graph connected graph consider construction convex copies of H Corollary covering critical set cycle dd(G decomposition defined denote diameter distance domination sequence distance graphs domination number double dominating set edge coloring edge covering element embedding entries exactly exists factors fc-dominates fc-stable set genus graph G graph of order Graph Theory Hence hole HSOLSSOM of type induced subgraph integer isolated vertices isomorphic Lemma Let G line graphs lower bound magic graph median graphs minimum MPT(Kv number of edges obtained pairs partial latin square partition path Proof Proposition quadratic residue region result segments square of order star graph step domination set subsets subsquare Suppose Theorem 2.1 thrackle total number tree unique completion upper bound vertex set VNI(G