Some Results On The Strong Trace Of Graphs

Biopolymer is the most basic part of life.For example:polypeptides,those are natural polymers,which can be self-assembled into ordered nanostructures.The stable trace and strong trace were introduced in[Sandi Klavzar,Jernej Rus,Stable traces as a model for self-assembly of polypeptide nanoscale polyhedrons,MATCH Commun.Math.Comput.Chem.70(2013)317-330.]and[Gasper Fijavz,Tomaz Pisanski,Jernej Rus,Strong traces model of selfassembly of polypeptide structures,MATCH Commun.Math.Comput.Chem.71(2014)199-212.],respectively,to provide the underlying mathematical model for self-assembly of polypeptide nanostructures.In this paper,we introduce the E1-antiparallel strong trace of a graph G =(V,E),which is a strong trace of G with only the edges of E1(?)E are traversed in the opposite direction.Given a graph G and its edge subset E1,the main purpose of the paper is to consider under which conditions G has an E1-antiparallel strong trace,and we solved it in the case that E1 is an independent edge set,E1 induces a path and forest of G,respectively.according to the existence theorem of E-restricted strong trace,we designed an algorithm to calculate the number of E-restricted strong trace of polyhedrons.we defined the Anti(G)and proved that the antiparallel edges of E-restricted strong trace with the smallest number of antiparallel edges induce a forest and designed an algorithm to calculate the Anti(G).