The Path-Structure Connectivity of Augmented k-ary n-cubes

For connected graphs G and H, the H-structure connectivity K (G; H) (resp. H-substructure connectivity K-s(G; H)) of G is the minimum cardinality of a set of subgraphs F of G such that each is isomorphic to H (resp. to a connected subgraph of H) so that G-F is disconnected or singleton. In this paper, we consider P-t-structure connectivity and P-t-substructure connectivity of augmented k-ary n- cubes AQ(n,k) forn >= 2, k >= 3 and 1 <= t <= 4n-2. We obtain that K (AQ(n,k); P-t) = K-s(AQ(n,k); P-t) = (t)/(4n-2) +1 for t | 4n - 2,t (sic) 2n - 1, t > 6, n >= 3 and k >= 4; K (AQ(n,k); P-t) = K-s(AQ(n,k); P-t) = [(t)/(4n-2)] , in other cases.