Super-minimally $3$-connected graphs

In this paper, we introduce super-minimally $k$-connected graphs, those $k$-connected graphs in which no proper subgraph is $k$-connected. For $k$ greater than or equal to three, this class lies strictly between the classes of minimally $k$-connected graphs and uniformly $k$-connected graphs. In particular, we determine the minimum number of degree-$3$ vertices in a super-minimally $3$-connected graph, thereby extending a result of Halin on minimally $3$-connected graphs. In addition, we determine the maximum number of edges in a super-minimally $3$-connected graph, extending Xu's result for uniformly $3$-connected graphs, and providing an analogue of Halin's result for minimally $3$-connected graphs.