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adjacency matrix adjacent algorithm antilattice assume automorphism bipartite graph blocks cardinality chromatic clique color complete graph component compute consider construction contains continued fraction contradiction cordial labeling Corollary CPij cyclic edge dacard defined definition degree delete denote digraph disjoint distance dominating set domination number edge connectivity edge-connected edge-optimal edges receive elements exists finite graph G Graph Theory Helly property Hence hyperedges hypergraph induced subgraph integer intersection isolated vertices isometrically embedded isomorphic least Lemma Let G linear magic squares Mathematics matroid matroid basis clutter maximal-clique partition minimal multigraphical node nonempty obtained odd cycle one-sided dominating set orthogonal polygons outerplanar graphs pair path algebra paths of type permutation planar graph plane point-distinguishing points polynomial positive integer Proof Proposition prove receive the label result satisfies sequence shortest square lattice staircase n-paths Subcase subgroup subset super edge-magic Suppose Theorem total dominating tree vector vertex set