Figure 8: Rollback spirals.
of times it “winds around” $P$ inside the ring (see Fig. 9). At least intuitively, it should be clear that winding numbers and non-rollback spirals are related. In particular, each ring can only have $2^{\mathcal{O}(k)}$ visitors and crossings subpaths (because the size of each separator is $2^{\mathcal{O}(k)}$), and we only have $\mathcal{O}(k)$ rings to deal with. Thus, it is possible to show that if the winding number of every crossing subpath is upper bounded by $2^{\mathcal{O}(k)}$, then the total number of non-rollback spirals is upper bounded by $2^{\mathcal{O}(k)}$ as well. The main tool we employ to bound the winding number of every crossing path is the following known result (rephrased to simplify the overview).
Proposition 2.2 ([14]). Let $G$ be a graph embedded in a ring with a crossing path $P$. Let $\mathcal{P}$ and $Q$ be two collections of vertex-disjoint crossings paths of the same size. (A path in $\mathcal{P}$ can intersect a path in $Q$, but not another path in $\mathcal{P}$.) Then, $G$ has a collection of crossing paths $\mathcal{P}'$ such that (i) for every path in $\mathcal{P}$, there is a path in $\mathcal{P}'$ with the same endpoints and vice versa, and (ii) the maximum difference between (the absolute value of) the winding numbers with respect to $P$ of any path in $\mathcal{P}$ and any path in $Q$ is at most 6.
To see the utility of Proposition 2.2, suppose momentarily that none of our rings has visitors. Then, if we could ensure that for each of our rings, there is a collection $Q$ of vertex-disjoint paths of maximum size such that the winding number of each path in $Q$ is a constant, Proposition 2.2 would have the following implication: if there is a solution, then we can modify it within each ring to obtain another solution such that each crossing subpath of each of its paths will have a constant winding number (under the supposition that the rings are disjoint, which we will deal with later in the overview), see Fig. 9. Our situation is more complicated due to the existence of visitors—we need to ensure that the replacement $\mathcal{P}'$ does not intersect them. On a high-level, this situation is dealt with by first showing how to ensure that visitors do not “go too deep” into the ring on either side of it. Then, we consider an “inner ring” where visitors do not exist, on which we can apply Proposition 2.2. Afterwards, we are able to bound the winding number of each crossing path by $2^{\mathcal{O}(k)}$ (but not by a constant) in the (normal) ring.
Modifying $R$ within Rings. To ensure the existence of the aforementioned collection $Q$ for each ring, we need to modify $R$. To this end, consider a long path $P$ with separators $S_u$ and $S_v$, and let $\mathcal{P}'$ be the subpath of $P$ inside the ring defined by the two separators. We compute a maximum-sized collection of vertex-disjoint paths $\text{Flow}(u, v)$ such that each of them has one endpoint in $S_u$ and the other in $S_v$.³ Then, we prove a result that roughly states the following.
Lemma 2.2. There is a path $\mathcal{P}^*$ in the ring defined by $S_u$ and $S_v$ with the same endpoints as $\mathcal{P}'$ crossing each path in $\text{Flow}(u, v)$ at most once. Moreover, $\mathcal{P}^*$ is computable in linear time.
³This flow has an additional property: there is a tight collection of $\mathcal{C}(u, v)$ of concentric cycles separating $S_u$ and $S_v$ such that paths in $\text{Flow}(u, v)$ do not “oscillate” too much between any two cycles in the collection. Such a maximum flow is said to be minimal with respect to $\mathcal{C}(u, v)$.