{2, 3} -Extraconnectivities of hypercube-like networks

A subset of vertices X is said to be a cutset if G-X is not connected. A cutset X is called an ^Rg-cutset if every component of G-X has at least g+1 vertices. If G has at least one ^Rg-cutset, the g-extraconnectivity of G is then defined as the minimum cardinality over all ^Rg-cutsets of G. In this paper, we first show that the 2-extraconnectivity of an n-dimensional hypercube-like network is 3n-5 for n≥5. This improves on the previously best known result, which showed that the 2-extraconnectivity of an n-dimensional hypercube-like network is 3n-5 for n≥8. We further demonstrate that the 3-extraconnectivity of an n-dimensional hypercube-like network is 4n-9 for n≥6. Based on the above results, the 2-extraconnectivity and 3-extraconnectivity of several interconnection networks, including hypercubes, twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes, generalized twisted cubes, recursive circulants, and Mcubes, can be determined efficiently.