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Fix a finite undirected graph H and two bi-infinite walks x and y in H. We say that x and y are adjacent if the ith step of x is adjacent to the ith step of y. x and y are said to be parallel if there exists a sequence x=x^1, x^2, …, x^n=y of paths such that x^j is adjacent to x^{j+1} for all j. Here the minimum number of steps n is called the parallelism distance between x and y. We wish to address the following question: When is the parallelism distance between paths in a graph bounded above? While we are not able to fully address the question, we do record some partial progress in this direction.
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