For a connected graph G, the r th extraconnectivity ¿r(G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r+1 vertices. The standard connectivity and superconnectivity correspond to ¿0(G) and ¿1(G), respectively. The minimum r-tree degree of G , denoted by ¿r(G), is the minimum cardinality of N(T) taken over all trees T¿G of order |V(T)|=r+1, N(T) being the set of vertices not in T that are neig...
For a connected graph G, the r th extraconnectivity ¿r(G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r+1 vertices. The standard connectivity and superconnectivity correspond to ¿0(G) and ¿1(G), respectively. The minimum r-tree degree of G , denoted by ¿r(G), is the minimum cardinality of N(T) taken over all trees T¿G of order |V(T)|=r+1, N(T) being the set of vertices not in T that are neighbors of some vertex of T . When r=1, any such considered tree is just an edge of G . Then, ¿1(G) is equal to the so-called minimum edge-degree of G , defined as ¿(G)=min{d(u)+d(v)-2:uv¿E(G)}, where d(u) stands for the degree of vertex u. A graph G is said to be optimally r -extraconnected, for short ¿r-optimal, if ¿r(G)¿¿r(G). In this paper, we present some sufficient conditions that guarantee ¿r(G)¿¿r(G) for r¿2. These results improve some previous related ones, and can be seen as a complement of some others which were obtained by the authors for r=1