Local connectivity, local degree conditions, some forbidden induced subgraphs, and cycle extendability

In particular, for a connected, locally connected graph G of order at least 3, our results are as follows: If G is (K1+(K1∪K2))-free, then G is weakly pancyclic. If G is (K1+(K1∪K2))-free, then G is fully cycle extendable if and only if 2δ(G)≥n(G). If G is {K1+K1+K̄3,K1+P4}-free or {K1+K1+K̄3,K1+(K1∪P3)}-free, then G is fully cycle extendable. If G is distinct from K1+K1+K̄3 and {K1+P4,K1,4,K2+(K1∪K2)}-free, then G is fully cycle extendable. Furthermore, we prove that a degree condition weaker than locally Dirac or locally Ore guarantees fully cycle extendability.