3-restricted connectivity of graphs with given girth

Let G = (V,E) be a connected graph. X ⊂ V (G) is a vertex set. X is a 3-restricted cut of G, if G-X is not connected and every component of G-X has at least three vertices. The 3-restricted connectivity κ3(G) (in short κ3) of G is the cardinality of a minimum 3-restricted cut of G. X is called κ3-cut, if |X| = κ3. A graph G is κ3-connected, if a 3-restricted cut exists. Let G be a graph girth g ≥ 4, ξ3(G) is min{d(x) + d(y) + d(z) − 4: xyz is a 2-path of G}. It will be shown that κ3(G) = ξ3(G) under the condition of girth.