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The graph resulting from contracting edge \( e \) is denoted \( G/e \). An edge \( e \) is radius-essential if \( rad(G/e) < rad(G) \). Let \( c_r(G) \) denote the number of radius-essential edges in graph \( G \). In this paper, we study realizability questions relating to the number of radius-essential edges, give bounds on \( c_r(G) \) in terms of radius and order, and we characterize various classes of graphs achieving extreme values of \( c_r(G) \).