  Let G be a graph. Consider two nonadjacent vertices x and y that have a
common neighbor. Folding G with respect to x and y is the operation which
identifies x and y. After a maximal series of foldings the graph is a disjoint
union of cliques. The minimal clique number that can appear after a maximal
series of foldings is equal to the chromatic number of G. In this paper we
consider the problem to determine the maximal clique number which can appear
after a maximal series of foldings. We denote this number as Sigma(G) and we
call it the max-folding number. We show that the problem is NP-complete, even
when restricted to classes such as trivially perfect graphs, cobipartite graphs
and planar graphs. We show that the max-folding number of trees is two.
