The IH-complex of spatial trivalent graphs
The authors define the IH-complex on the set of spatial trivalent graphs as follows: The vertices of the complex are isotopy classes of spatial trivalent graphs, and two vertices are connected by an edge if one graph can be obtained from another by an IH-move. An IH-move can be described as follows: begin with two adjacent vertices, \(v\) and \(w\), and suppose \(v\) is also adjacent to \(a\) and \(b\), while \(w\) is adjacent to \(x\) and \(y\). Then replace \(v\) and \(w\) by adjacent vertices \(v'\) and \(w'\) such that \(v'\) is adjacent to \(a\) and \(x\), and \(w'\) is adjacent to \(b\) and \(y\). The distance between two vertices in the complex is the minimal number of IH-moves required to transform one graph into the other (or \(\infty\) if there is no such sequence of moves). The authors prove that the trivial spatial theta graph and the trivial spatial handcuff graph are adjacent in the complex, but while the degree of the handcuff graph is 1, the degree of the theta graph is \(\infty\) (so the complex is not locally finite). They go on to define flows along the graphs, and use these to provide a lower bound on the distance between two spatial graphs in the complex. The also construct a family of spatial graphs which realize all integral distances between vertices in the complex.
scientific article; zbMATH DE number 5869980
scientific article