Consider a directed graph G and the (generally infinite) set Walks(G) of the walks in that graph. We are interested in ordering Walks(G) as w1, w2,... in a way that ensures that each walk is obtained from the previous one by a bounded number of elementary operations, namely, push operations (which add a new edge at either end of the walk) and pop operations (which remove the first or last edge of the walk). Our goal is to optimize the aforementioned bound, when it exists, in particular when imposing the additional requirement that walks must be ordered in increasing order of length. To this end, we introduce a condition on graphs inspired by earlier work, called inner-connectedness, which is testable in linear time; we show that inner-connected graphs are precisely those which admit bounded walk orderings. Focusing on such graphs, we then introduce and study the pp-rank and the pp-increasing-rank. Namely, the pp-rank of a graph is the best possible bound on the number of push-pop operations in a bounded walk ordering for that graph; and the pp-increasing-rank is the same where we further impose that shorter walks appear before longer walks in the ordering. We show that the pp-rank of inner-connected graphs is in fact always bounded by 3. The pp-increasing-rank of inner-connected graphs is also always finite, but it can be up to linear in the input graph. More precisely, we show that the pp-increasing-rank is related to a new parameter we introduce, the peripherality of graphs. In particular, for cyclic graphs, peripherality is related to the pp-increasing-rank up to multiplicative and additive constants. We last study how to compute these parameters, and show that while peripherality can be computed in polynomial time, the pp-increasing-rank itself is generally NP-hard to compute.

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