Graphs and Trees

1. Graphs and Trees • This handout: • Total degree of a graph • Applications of Graphs

2. Graph properties • Definition: The total degree of a graph is the sum of the degrees of all its nodes. • Theorem: If G is any graph, then the total degree of G equals twice the number of edges of G: the total degree of G = 2 (the number of edges of G) • Corollary 1: The total degree of a graph is even. • Corollary 2: In any graph there are an even number of vertices of odd degree. • Application to an Acquaintance Graph: Is it possible in a group of five people for each to be friends with exactly three others?

3. Terminology of Graph: Paths • A path between two nodes is a sequence of distinct nodes and edges connecting these nodes. Example: • Two nodes are calledconnected if there is a path between them. • Fact: For any two nodes a and b of a graph, there is an efficient way to determine whether a and b are connected or not. a b

4. An application of graphs in solving a puzzle • From an initial position on the left bank of a river, a ferryman wants to transport a wolf, a goat, and a cabbage to the right bank. Ferryman’s boat is only big enough to transport one object at a time, other than himself. For obvious reasons, • the wolf cannot be left alone with the goat; • the goat cannot be left alone with the cabbage. • How should the ferryman proceed?

5. An application of graphs in solving a puzzle To solve the puzzle, create the following graph: • Create a node for each allowable arrangement. E.g., ( fg | wc ) is an allowable arrangement since the ferryman and the goat are on the left bank, and the wolf and the cabbage are on the right bank. • Create an edge between two nodes if it is possible to go from the arrangement of one node to the arrangement of the other node by a single ferry trip. E.g., there is an arc between nodes ( fgw | c ) and ( w | fgc ) because the transition from the first node to the second node can be realized by a single trip of the ferryman with the goat from the left bank to the right bank.

6. An application of graphs in solving a puzzle fwgc | fwg | c fwc | g fgc | w fg | wc The resulting graph is: To transport everything from the left bank to the right bank, we need to find a path from node ( fwgc | ) to node ( | fwgc ) in the graph. There are two this kind of paths. One of them: (fwgc | )  (wc | fg)  (fwc | g)  (w | fgc)  (fwg | c)  (g | fwc)  (fg | wc)  ( | fwgc) c | fwg | fwgc wc | fg w | fgc g | fwc