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  • #MA284 - Discrete Mathematics
  • Previous Topic: Colouring Graphs; Eulerian & Hamiltonian Graphs
  • Next Topic: Matrices
  • Relevant Slides: MA284-Week11.pdf

  • Trees

    • What is an acyclic graph or a forest? #card card-last-interval:: 2.8 card-repeats:: 1 card-ease-factor:: 2.6 card-next-schedule:: 2022-11-19T16:18:08.222Z card-last-reviewed:: 2022-11-16T21:18:08.222Z card-last-score:: 5
      • A graph that has no circuits is called acyclic or a "forest".
      • It is made up of trees.
    • What is a tree? #card card-last-interval:: 0.98 card-repeats:: 1 card-ease-factor:: 2.36 card-next-schedule:: 2022-11-17T20:21:51.032Z card-last-reviewed:: 2022-11-16T21:21:51.032Z card-last-score:: 3
      • A tree is a connected, acyclic graph.
      • A graph is a tree if and only if there is a unique path between any two vertices.
        • If a graph has a unique path between pairs of vertices then it has no cycles, but there is a path between each pair, so it is connected.
      • A tree is a forest.

    • If T is a tree, then e = v - 1. #card

      card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-18T00:00:00.000Z card-last-reviewed:: 2022-11-17T09:45:56.481Z card-last-score:: 1
      • If T is a tree (i.e., a connected acyclic graph) with v vertices, then it has v-1 edges.
      • The converse of this statement is also true. ((6374e0df-7be4-4928-bf03-c336e517bbf9))
    • It can be difficult to determine if a very large graph is a tree just by inspection.
      • If we know that it has no cycles, then we need to verify that it is connected. The following result (the converse of the previous one) can be useful:

        • If e = v-1, then T is a tree. #card

          id:: 6374e0df-7be4-4928-bf03-c336e517bbf9 card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-18T00:00:00.000Z card-last-reviewed:: 2022-11-17T09:46:20.887Z card-last-score:: 1
          • If a graph with v vertices has no cycles, and has e = v-1 edges, then it is a tree.

  • Applications

    • What is a Decision Tree? #card card-last-interval:: 0.98 card-repeats:: 1 card-ease-factor:: 2.36 card-next-schedule:: 2022-11-18T08:46:46.255Z card-last-reviewed:: 2022-11-17T09:46:46.255Z card-last-score:: 3
      • A Decision Tree is a graph where each node represents a possibility, and each branch/edge from that node is a possible outcome.

  • Spanning Trees

    • What is a Spanning Tree? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-18T00:00:00.000Z card-last-reviewed:: 2022-11-17T09:47:19.672Z card-last-score:: 1
      • Given a (simple) graph G, a spanning tree of G is a subgraph of G that is a tree, and contains every vertex of G.
    • Lots of spanning trees are possible, and there are numerous ways of finding them. Here are two:

      • Algorithm 1

        • (i) Identify a cycle in the graph. (ii) Delete an edge in that cycle, taking care not to disconnect the graph. (iii) Keep going until all cycles have been removed.

      • Algorithm 2

        • (i) Start with just the vertices of the graph (no edges). (ii) Add an edge from the original graph, as long as it does not form a cycle. (iii) Stop when the graph is connected.
    • For many applications, we need to consider a wider class of graphs: weighted graphs.

    • Minimum Spanning Trees

      • What is a minimum spanning tree? #card card-last-interval:: 2.8 card-repeats:: 1 card-ease-factor:: 2.6 card-next-schedule:: 2022-11-21T13:33:14.273Z card-last-reviewed:: 2022-11-18T18:33:14.274Z card-last-score:: 5
        • A minimum spanning tree is a spanning tree with the minimum possible total edge weight.

      • Kruskal's Algorithm

        • (i) Start with just the vertices. (ii) Add the edge with the least weight that does not form a cycle. (iii) Keep going until the graph is connected.

      • Prim's Algorithm

        • (i) Choose any vertex from the original graph. (ii) Add the edge incident to any vertex in the connected component with least weight that has the least weight and does not create a cycle. (iii) Stop when you reach all the vertices of the original graph.