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Graph Theory: Chains (Paths) and Cycles (Circuits)

Graph Theory: Chains (Paths) and Cycles (Circuits). Even more terminology. Chain (Path): Sequence of edges where you do not start and end at the same vertex Simple chain : Does not repeat an edge Length : number of edges in the chain

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Graph Theory: Chains (Paths) and Cycles (Circuits)

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  1. Graph Theory: Chains (Paths) and Cycles (Circuits)

  2. Even more terminology • Chain (Path): Sequence of edges where you do not start and end at the same vertex • Simple chain: Does not repeat an edge • Length: number of edges in the chain • Distance: written as d(A, B) and is equal to the shortest length connecting two vertices • Cycle (Circuit): A chain beginning and ending at the same vertex • Simple cycle: does not pass through the same edge more than once.

  3. Ex C. B. D. .E A.

  4. Chain: ABCDE, ABCDC, ABCE • Simple chain: ABCDE, ABCE • Length: ABCDE = 4, ABCDC = 4, ABCE = 3 • Distance (A, E) = 2 • Cycle: ABCDA, ABCDECDA,DCED • Simple cycle: ABCDA, DCED

  5. Euler Chain and Cycle Euler Chain: A chain that passes through each vertex once and only once (do not end where you start) Euler Cycle: A cycle that passes through each vertex once and only once (must end where you start)

  6. Ex Euler Chain: BACDBC A. B. .C .D

  7. Ex Euler Cycle: ACDBA A. B. .C .D

  8. How to tell if there is a Euler Chain or Cycle • If each vertex has an even degree = Euler cycle • If there are exactly 2 vertices with odd degree = Euler chain

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