Graph Theory

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Trees

For an unconnected graph G, the following are equivalent

  • G is connected and acyclic (contains no cycles).
  • G is acyclic, and a simple cycle is formed if any edge is added to G.
  • G is connected, but would become disconnected if any single edge is removed from G.
  • G is connected and the 3-vertex complete graph K3 is not a minor of G.
  • Any two vertices in G can be connected by a unique simple path.