令(4)→(5): T is a graph with no simple circuit, and if x and y are nonadjacent vertices of T, then T+x, y contains exactly a simple circuit. then T-x, yi is disconnected d We prove T is connecte ed and if{x,y}∈E(T) o Suppose T is disconnected. There are vertices Vi and j Such tI that there is not any simple path between vi and v Add an edge{y;}→T %o The new graph has also no simple circuit Contradiction❖ (4)→(5): T is a graph with no simple circuit, and if x and y are nonadjacent vertices of T, then T+{x,y} contains exactly a simple circuit. We prove “T is connected and if {x,y}E(T) then T-{x,y} is disconnected”. ❖ Suppose T is disconnected. There are vertices vi and vj such that there is not any simple path between vi and vj . ❖ Add an edge {vi ,vj } → T ❖ The new graph has also no simple circuit. Contradiction