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A convex subgraph of a connected graph G is a subgraph of G induced by a convex subset of V (G). For a proper convex subgraph H of G, we say that G is H-covex k-accessible if for every vertex v ∈ V (G) \ V (H), the distance between v and H in G is at most k. The H-convex accessibility number of G is the minimum k for which G is Hconvex k-accessible. In this paper, we established sharp bounds for the H-convex accessibility number of graphs and characterized graphs with H-convex accessibility number equal to some positive integer. Moreover, we established results on the H-convex accessibility numbers of some special graphs.