Weak Graph Map Homotopy and Its Applications

The authors introduce a notion of a weak graph map homotopy (they call it M-homotopy), discuss its properties and applications. They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001. The difference between them is that the weak graph map homotopy transformation is defined in terms of maps, while the graph homotopy transformation is defined by means of combinatorial operations. They discuss its advantages over the graph homotopy transformation. As its applications, they investigate the mapping class group of a graph and the 1-order M P-homotopy group of a pointed simple graph. Moreover, they show that the 1-order M P-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.