On $k$-Connected $\mathrm{\Gamma }$-Extensions of Binary Matroids

Slater introduced the point-addition operation on graphs to classify 4-connected graphs. The $\mathrm{\Gamma }$-extension operation on binary matroids is a generalization of the point-addition operation. In this paper, we obtain necessary and sufficient conditions to preserve $k$-connectedness of a binary matroid under the $\mathrm{\Gamma }$-extension operation. We also obtain a necessary and sufficient condition to get a connected matroid from a disconnected binary matroid using the $\mathrm{\Gamma }$-extension operation.