On Domination and Digital Convexity Parameters

Suppose V is a finite set and C a collection of subsets of V that contains ∅ and V and is closed under taking intersections. Then the ordered pair (V, C) is called a convexity and the elements of C are referred to as convex sets. For a set S ⊆ V , the convex hull of S relative to C, denoted by CHC(S), is the smallest convex set containing S. The Caratheodory number, relative to a given convexity, is the smallest integer c such that for any subset S of V and any point v ∈ CHC(S) there is a subset F of S with |F | ≤ c such that v ∈ CHC(F ). A subset X of V is said to admit a Radon partition if X can be partitioned into two sets X1, X2 such that CHC(X1) ∩ CHC(X2) ̸= ∅. 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 ∈ V , N [v] ⊆ N [S] implies v ∈ S. A set X is irredundant if N [x]−N [X−{x}] ̸= ∅ for all x ∈ 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 dominating set for a vertex v of G, if N [v] ∈ N [X]. The upper local domination number of v, denoted by lΓ(v), is the maximum cardinality of a minimal local dominating set for v. The upper local domination number of a graph G, denoted by lΓ(G), is defined as lΓ(G) = max{lΓ(v) | v ∈ V }. We show that for the digital convexity of a graph G: (i) the Caratheodory number equals lΓ(G) and (ii) the Radon number is bounded above by IR(G)+1 and below by β(G)+1 where β(G) is the independence ∗Supported by an NSERC grant CANADA.