MT4514: Graph Theory

Graph Theory 451. The vertices of degree 1 are vertices 3, 5, 6 and 8, the one with the smallestlabel is vertex 3.2. The vertex adjacent to vertex 3 is vertex 2, so the sequence starts with 2.3. Removing vertex 3 and its incident edge gives the following tree:4. The vertices of degree 1 are vertices 5, 6 and 8, the smallest label is vertex 5.5. The vertex adjacent to vertex 5 is vertex 2, so the next number in the sequenceis 2.6. Removing vertex 5 and its incident edge gives the following tree:7. The vertices of degree 1 are vertices 2, 6 and 8, the smallest label is vertex 2.8. The vertex adjacent to vertex 2 is vertex 4, so the next number in the sequenceis 4.9. Removing vertex 2 and its incident edge gives the following tree:10. Continue in the way, removing the edges (6, 1), (1, 4), (4, 7) to get the Prüfersequence (2, 2, 4, 1, 4, 7).In order to reverse the operation, we take a Prüfer sequence and apply thefollowing three steps:1. Draw the n vertices, labelling them from 1 to n, and make a list L of thenumbers from 1 to n.