Forbidden subgraphs and hamiitonian properties in the square of a connected graph

Various Harniltonian-like properties are investigated in the squares of connected graphs free of some set of forbidden subgraphs. The star K_1,4 the subdivision graph of K_1,3, and the subdivision graph of K_1,3 minus an endvertex play central roles. In particular, we show that connected graphs free of the subdivision graph of K_1,3 minus an endvertex have vertex pancyclic squares.