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A dominating set of a graph \( G \) is a set of vertices \( D \) such that for all \( \nu \in V(G) \), either \( \nu \in D \) or \( [\nu,d] \in E(G) \) for some \( d \in D \). The cardinality-redundance of a vertex set \( S \), \( CR(S) \), is the number of vertices \( x \in V(G) \) such that \( |N[x] \cap S| \geq 2 \). The cardinality-redundance of \( G \) is the minimum of \( CR(S) \) taken over all dominating sets \( S \). A set of vertices \( S \) such that \( CR(S) = CR(G) \) is a \( \gamma_{CR} \)-set, and the size of a minimum \( \gamma_{CR} \)-set is denoted \( \gamma_{CR}(G) \).
Here, we are concerned with extremal problems concerning cardinality-redundance. We give the maximum number of edges in a graph with a given number of vertices and given cardinality-redundance. In the cases that \( CR(G) = 0 \) or \( 1 \), we give the minimum and maximum number of edges of graphs when \( \gamma_{CR}(G) \) is fixed, and when \( CR(G) = 2 \), we give the maximum number of edges of graphs where \( \gamma_{CR}(G) \) is fixed. We give the minimum and maximum values of \( \gamma_{CR}(G) \) when the number of edges are fixed and \( CR(G) = 0, 1 \), and we give the maximum values of \( \gamma_{CR}(G) \) when the number of edges are fixed and \( CR(G) = 2 \).