Suppose \( V \) is a finite set and \( \mathcal{C} \) a collection of subsets of \( V \) that contains \( \emptyset \) and \( V \) and is closed under taking intersections. Then the ordered pair \( (V, \mathcal{C}) \) is called a \({convexity}\) and the elements of \( \mathcal{C} \) are referred to as \({convex\; sets}\). For a set \( S \subseteq V \), the \({convex\; hull}\) of \( S \) relative to \( \mathcal{C} \), denoted by \( CH_{\mathcal{C}}(S) \), is the smallest convex set containing \( S \). The \({Carathéodory\; number}\), relative to a given convexity, is the smallest integer \( c \) such that for any subset \( S \) of \( V \) and any point \( v \in CH_{\mathcal{C}}(S) \), there is a subset \( F \) of \( S \) with \( |F| \leq c \) such that \( v \in CH_{\mathcal{C}}(F) \). A subset \( X \) of \( V \) is said to admit a \({Radon \;partition}\) if \( X \) can be partitioned into two sets \( X_1 \) and \( X_2 \) such that \( CH_{\mathcal{C}}(X_1) \cap CH_{\mathcal{C}}(X_2) \neq \emptyset \). The \({Radon\; number}\) of a convexity is the smallest integer \( r \) (if it exists) such that every subset \( X \) of \( V \) with at least \( r \) elements admits a Radon partition.
A set \( S \) of vertices in a graph \( G \) with vertex set \( V \) is \({digitally}\) convex if for every vertex \( v \in V \), \( N[v] \subseteq N[S] \) implies \( v \in S \). A set \( X \) is \({irredundant}\) if \( N[X] – N[X – \{x\}] \neq \emptyset \) for all \( x \in X \). The maximum cardinality of an irredundant set is the \({upper\; irredundance\; number}\) of \( G \), denoted by \( IR(G) \). A set \( X \) of vertices in a graph \( G \) is a \({local\; irredundant}\) set for a vertex \( v \) of \( G \), if for each \( x \in X \), \( x \in N[v] – N[X – \{x\}] \) or \( x \) is adjacent to a vertex of \( N[v] – N[X – \{x\}] \). The \({upper\; local \;irredundance \;number}\) of \( v \), denoted by \( l_{IR}(v) \), is the maximum cardinality of a local irredundant set for \( v \). The \({upper\; local\; irredundance\; number}\) of a graph \( G \), denoted by \( l_{IR}(G) \), is defined as \( l_{IR}(G) = \max \{ l_{IR}(v) \mid v \in V \} \).
We show that for the digital convexity of a graph \( G \):
(i) The Carathéodory number equals \( l_{IR}(G) \).
(ii) The Radon number is bounded above by \( IR(G) + 1 \) and below by \( \beta(G) + 1 \) where \( \beta(G) \) is the independence number of \( G \). For the latter result, it is shown that there are classes of graphs for which the lower (respectively, upper) bound is attained, while the difference between the upper irredundance number and the independence number can be made as large as we wish. Moreover, there are graphs for which the Radon number of the digital convexity lies strictly between the bounds given in (ii) and does not equal one more than the upper domination number.