Refining a chain theorem from matroids to internally 4-connected graphs

Graph theory and matroid theory are interconnected with matroids providing a way to generalize and analyze the structural and independence properties within graphs. Chain theorems, vital tools in both matroid and graph theory, enable the analysis of matroid structures associated with graphs. In a significant contribution, Chun, Mayhew, and Oxley [2] established a chain theorem for internally 4-connected binary matroids, clarifying the operations involved. Our research builds upon this by specifying the matroid result to internally 4-connected graphs. The primary goal of our research is to refine this chain theorem for matroids into a chain theorem for internally 4-connected graphs, making it more accessible to individuals less acquainted with matroid theory. • We establish the equivalence of internally 4-connectivity between matroids and graphs. • We refine the chain theorem on binary, internally 4-connected matroids to derive operations generating such a graph. • We introduce 19 operations integral to deriving our main theorem.