Joint-Tree Model and the Maximum Genus of Graphs

The vertex v of a graph G is called a 1-critical-vertex for the maximum genus of the graph, or for simplicity called 1-critical-vertex, if G v is a connected graph and M(G v) = M(G) 1. In this paper, through the joint-tree model, we obtained some types of 1-critical-vertex, and get the upper embeddability of the Spiral S n.