Question: Computer Science: Give a minimum-width tree decomposition for the following graph. Determine whether the following graph families are closed under the minor order for any

Computer Science:

Give a minimum-width tree decomposition for the following graph. Determine whether the following graph families are closed under the minor order for any fixed k. If the family is closed, provide justification. If the family is not closed, give a counterexample. Graphs that contain a path of length k. Graphs that do not contain a path of length k. Graphs that contain a dominating set of size k. Graphs that do not contain a dominating set of size k. Graphs that contain a k-vertex cycle Graphs that do not contain a k-vertex cycle. Give a minimum-width tree decomposition for the following graph. Determine whether the following graph families are closed under the minor order for any fixed k. If the family is closed, provide justification. If the family is not closed, give a counterexample. Graphs that contain a path of length k. Graphs that do not contain a path of length k. Graphs that contain a dominating set of size k. Graphs that do not contain a dominating set of size k. Graphs that contain a k-vertex cycle Graphs that do not contain a k-vertex cycle

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