# On $k$-Connected $\Gamma$-Extensions of Binary Matroids

## TL;DR

This paper characterizes when the $ ext{Gamma}$-extension operation preserves $k$-connectedness in binary matroids and provides conditions to connect disconnected matroids through this operation.

## Contribution

It offers necessary and sufficient conditions for preserving $k$-connectedness and connectivity in binary matroids under the $ ext{Gamma}$-extension operation.

## Key findings

- Conditions for $k$-connectedness preservation
- Criteria for connecting disconnected matroids
- Generalization of point-addition in graphs

## Abstract

Slater introduced the point-addition operation on graphs to classify 4-connected graphs. The $\Gamma$-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 $\Gamma$-extension operation. We also obtain a necessary and sufficient condition to get a connected matroid from a disconnected binary matroid using the $\Gamma$-extension operation.