Math 160, Chapter 7, Spanning trees Definition 1.1. a) A network is the same thing as a connected graph. b) A subgraph of a graph G is a graph whose edges and vertices are a subset of the edges and vertices of G. c) A tree is a network with no circuits (no closed paths). d) A spanning tree of a network is a subgraph that connects all the vertices and has no circuits. In particular it is a tree. e) A minimum spanning tree (MST) for a given network is the spanning tree of least total weight. Note. Important properties of a tree: 1) For any two vertices in a tree there is exactly one path joining them. 2) Every edge of a tree is a bridge. 3) A tree with n vertices has n − 1 edges. Kruskal’s Algorithm: For obtaining a minimum spanning tree. 1. Start with the edge of minimum weight. 2. From the remaining edges, select the edge of minimum weight such that no circuit is formed. 3. Repeat the process until you have a spanning tree. Note. Kruskal’s algorithm always gives a minimum spanning tree. Definition 1.2. a) A junction point of a network is a point where two or more branches (edges) of the network come together. b) An interior junction point is a junction point added to the original graph. c) A Steiner point is an interior junction point with three branches (edges) coming together at 120 degree angles. d) A network for a set of points is a connected graph joining all of the vertices that may also include additional vertices. e) A Steiner tree is a tree such that all interior junction points are Steiner points. Theorem 1.1. Interior Junction Rule for Shortest Networks. The shortest network connecting a set of points is either (i) a minimum spanning tree (no interior points) or (ii) a Steiner tree. Theorem 1.2. Shortest Networks for three vertices A, B, C. 1) If all of the angles in triangle ABC are less than 120 degrees then the shortest network is a Steiner tree obtained by placing a Steiner point inside the triangle. 2) If some angle is at least 120 degrees then the shortest network is the minimal spanning tree, that is, the two shortest edges.