Let G be a graph on vertex set [.], and for X C_ [.] let N(X) be the union of X and it's neighborhood in G. A family of sets ' C- 2 ["] is G-intersectin# if N(X) Y 0 for all X, Y '. In this paper we study the cardinality and structure of the largest k-uniform G-intersecting families on a fixed graph G
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