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2-spanner adjacent algorithm array assume automorphism bandwidth block Boolean cell chord clique codes color column combinatorial comparability graph complete mappings components consider construction contains contradiction Corollary corresponding cycle defined denote digraph disjoint maximum independent dominating set elements exists extend to disjoint face configuration fault pattern finite fractional coloring given graph G Graph Theory Halin graph Hamilton path Hamiltonian Hamiltonian path Hence induced subgraph isomorphic labels Lemma Let G linear lower bound Math matrix maximal maximum independent sets minimal minimum nodes NP-complete number of edges obtained optimal pair pairable partition permutation graph planar graph points polynomial positive integer problem Proof RCCC result schedule sequence sets in G shown in Figure spanning tree square strongly preserves subset subtree Suppose Theorem Traveling Classes tree spanner unate function unique vector vertex vertex of degree visibility graph weight