Giancarlo R., Mignosi F. (1994) Generalizations of the periodicity theorem of Fine and Wilf. In: Tison S. (eds) Trees in Algebra and Programming — CAAP'94. CAAP 1994. Lecture Notes in Computer Science, vol 787. Springer, Berlin, Heidelberg
We provide three generalizations to the two-dimensional case of the well known periodicity theorem by Fine and Wilf  for strings (the one-dimensional case). The first and the second generalizations can be further extended to hold in the more general setting of Cayley graphs of groups. Weak forms of two of our results have been developed for the design of efficient algorithms for two-dimensional pattern matching [2, 3, 6].