On Domination in Matroids of Graphs

The concept of domination has increasingly found powerful extensions within the structural framework of matroid theory. This paper investigates the structural, algebraic, and algorithmic properties of domination in matroids, unifying classical network domination with matroidal invariants such as closures, independent sets, and fundamental circuits. We formalise the notion of a dominating configuration in a general matroid M = (E,I) and analyse the combinatorial parameters governing its minimal representations. Specifically, we characterize the complexity of computing the minimum-weight domination number across prominent matroid classes, including graphic matroids. By leveraging the matroid rank function r(M), we establish novel lower and upper bounds for the domination invariant.