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An odd neighborhood transversal of a graph is a set of its vertices that intersects the set of neighbors of each of its vertices in an odd number of elements. In the case of grid graphs this odd number will be either one or three. We characterize those grid graphs that have odd neighborhood transversals.

Define a simple graph G to be k-superuniversal iff for any k-element simple graph K and for any full subgraph H of K every full embedding of H into G can be extended to a full embedding of K into G. We prove that for each positive integer k there exist finite k-superuniversal graphs, and e find upper and looer bounds on the smallest such graphs. We also… (More)

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