The study of a graph folding has a

The study of a graph folding has a long and distinguished history, with close
connections to important industrial applications. Linkage (planar graphs) folding
has applications in robotics and hydraulic tube bending. Paper folding has
applications in sheet metal bending, packaging, and air-bag folding[2] . Following
the great Soviet geometer A. Pogorelov[8] , M. El Naschie used the
folding to solve difficult problems related to shell structures in civil engineering
[7] .The meter stick can be regarded as a physical model of the path Pn.
After folding the meter stick, it becomes a physical model for the complete
graph K2. On the basis of this observation, the authors in[3] defined the folding
of a graph. Also they defined the folding number of a graph G by the
order of the smallest complete graph obtainable from G by a series of foldings.
David R. Wood[9] showed in unpublished work that the folding number of a
graph G equals the chromatic number of G. In [6], nada S. and Hamouda E.
introduced a new concept of the folding number of a graph G and computed
the number of folding maps f : Pn −→ Pn, where Pn is a path with n edges.
In this paper, the concept of based folding of a based graph is introduced and
the number of based folding F