On finding short reconfiguration sequences between independent sets

Assume we are given a graph $G$, two independent sets $S$ and $T$ in $G$ of size $k \geq 1$, and a positive integer $\ell \geq 1$. The goal is to decide whether there exists a sequence $\langle I_0, I_1, ..., I_\ell \rangle$ of independent sets such that for all $j \in \{0,\ldots,\ell-1\}$ the set $I_j$ is an independent set of size $k$, $I_0 = S$, $I_\ell = T$, and $I_{j+1}$ is obtained from $I_j$ by a predetermined reconfiguration rule. We consider two reconfiguration rules. Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the Token Sliding Optimization (TSO) problem asks whether there exists a sequence of at most $\ell$ steps that transforms $S$ into $T$, where at each step we are allowed to slide one token from a vertex to an unoccupied neighboring vertex. In the Token Jumping Optimization (TJO) problem, at each step, we are allowed to jump one token from a vertex to any other unoccupied vertex of the graph. Both TSO and TJO are known to be fixed-parameter tractable when parameterized by $\ell$ on nowhere dense classes of graphs. In this work, we show that both problems are fixed-parameter tractable for parameter $k + \ell + d$ on $d$-degenerate graphs as well as for parameter $|M| + \ell + \Delta$ on graphs having a modulator $M$ whose deletion leaves a graph of maximum degree $\Delta$. We complement these result by showing that for parameter $\ell$ alone both problems become W[1]-hard already on $2$-degenerate graphs. Our positive result makes use of the notion of independence covering families introduced by Lokshtanov et al. Finally, we show that using such families one can obtain a simpler and unified algorithm for the standard Token Jumping Reachability problem parameterized by $k$ on both degenerate and nowhere dense classes of graphs.


Introduction
tokens placed on the vertices of a graph such that no two tokens are placed on adjacent vertices. This gives rise to two natural adjacency relations between independent sets (or token configurations), also called reconfiguration steps. These reconfiguration steps, in turn, give rise to several combinatorial reconfiguration problems [28,27,8].
In the Token Sliding Reachability (TSR) problem, introduced by Hearn and Demaine [15], two independent sets are adjacent if one can be obtained from the other by removing a token from a vertex u and immediately placing it on another unoccupied vertex v with the requirement that {u, v} must be an edge of the graph. The token is said to slide from vertex u to vertex v along the edge {u, v}. Generally speaking, in the Token Sliding Reachability problem, we are given a graph G and two independent sets S and T of size k in G. The goal is to decide whether there exists a sequence of slides (a reconfiguration sequence) that transforms S to T . The TSR problem has been extensively studied [5,6,11,14,18,21,24]. It is known that the problem is PSPACE-complete, even on restricted graph classes such as planar graphs of bounded bandwidth (and hence pathwidth) [15,30,29], split graphs [3], and bipartite graphs [23]. However, Token Sliding Reachability can be decided in polynomial time on trees [11], interval graphs [5], bipartite permutation and bipartite distance-hereditary graphs [14], line graphs [17], and claw-free graphs [6]. In the Token Sliding Optimization (TSO) problem, we are additionally given a parameter ℓ and the goal is to decide if S can be transformed to T in at most ℓ token slides. Very little is known about the optimization variant of the problem other than the hardness results that follow immediately from the reachability variant. In fact, to the best of our knowledge, the only known polynomial-time solvable instances of TSO are those restricted to interval graphs [31,20], cographs [21], and spider trees (trees obtained by attaching paths to a central vertex) [16].
In the Token Jumping Reachability (TJR) problem, introduced by Kamiński et al. [21], we drop the restriction that the token should move along an edge of G and instead we allow it to move to any unoccupied vertex of G provided it does not break the independence of the set of tokens. That is, a single reconfiguration step consists of first removing a token on some vertex u and then immediately adding it back on any other unoccupied vertex v, as long as no two tokens become adjacent. The token is said to jump from vertex u to vertex v. Token Jumping Reachability is also PSPACE-complete on planar graphs of bounded bandwidth [15,30,29]. Lokshtanov and Mouawad [23] showed that, unlike Token Sliding Reachability, which is PSPACE-complete on bipartite graphs, the Token Jumping Reachability problem becomes NP-complete on bipartite graphs. On the positive side, it is "easy" to show that Token Jumping Reachability can be decided in polynomial-time on trees (and even on split/chordal graphs) since we can simply jump tokens to leaves (resp. vertices that only appear in the bag of a leaf in the clique tree) to transform one independent set into another. In the Token Jumping Optimization (TJO) problem, we are additionally given a parameter ℓ and the goal is to decide if S can be transformed to T in at most ℓ token jumps. To the best of our knowledge, the only known polynomial-time solvable instances of TJO are those restricted to even-hole-free graphs [21,26].
In this paper we focus on the parameterized complexity of the aforementioned problems with respect to parameters k and ℓ and when restricted to sparse classes of graphs. Given an NP-hard or PSPACE-hard problem, parameterized complexity [13] allows us to refine the notion of hardness; does the hardness come from the whole instance or from a small parameter? A problem Π is FPT (fixed-parameter tractable) parameterized by k if one can solve it in time f (k) · poly(n), for some computable function f (sometimes called FPT-time). In other words, the combinatorial explosion can be restricted to the parameter k. In the rest of the paper, we mainly consider parameters k (the number of tokens) and ℓ (the number of reconfiguration steps). TSO and TJO are known to be W[1]-hard (and XNL-complete [4]) parameterized by k + ℓ on general graphs [8]. TSR and TJR are known to be W[1]-hard (and XL-complete [4]) parameterized by k on general graphs [24] . When we restrict our attention to sparse classes of graphs, TSO and TJO are known to be fixed-parameter tractable when parameterized by ℓ on nowhere dense classes of graphs [26]. TJR and TJO are known to be fixed-parameter tractable parameterized by k on graphs of bounded degree [19]. For TJR, the problem becomes fixed-parameter tractable parameterized by k on biclique-free classes of graphs [7]. Finally, for TSR, the problem becomes fixed-parameter tractable parameterized by k on planar graphs, chordal graphs of bounded clique number, and graphs of bounded degree [2]. We refer the reader to the recent survey by Bousquet et al. [8] for more background on the parameterized complexity of these problems.
Given that TSO and TJO are fixed-parameter tractable when parameterized by ℓ on nowhere dense classes of graphs, it is natural to ask whether this result can be extended beyond nowhere dense graphs to biclique-free graphs. Even simpler, can we show that TSO and TJO remain fixed-parameter tractable when parameterized by ℓ on graph of bounded degeneracy? Recall that any degenerate or nowhere dense class of graphs is a biclique-free class, but not vice versa. Motivated by these questions, we show the following: Both problems are fixed-parameter tractable for parameter k + ℓ + d on d-degenerate graphs; Both problems are fixed-parameter tractable for parameter |N | + k + ℓ + d on graphs having a modulator N whose deletion leaves a d-degenerate graph (assuming N is given as part of the input); and Both problems are fixed-parameter tractable for parameter |M | + ℓ + ∆ on graphs having a modulator M whose deletion leaves a graph of maximum degree ∆. We complement these result by showing that for parameter ℓ alone both problems become W[1]-hard already on 2-degenerate graphs.
In fact, our hardness reductions construct 2-degenerate graphs which can be partitioned into two sets V 1 and V 2 , where V 1 is an independent set and every vertex in V 2 has constant degree in the graph. Hence, our positive result for parameter |M |+ℓ+∆ shows that when |M | is part of our parameter we can drop k and still obtain fixed-parameter tractable algorithms; and when |M | (and k) is not part of the parameter the problem is W[1]-hard.
Most of our positive results make use of the notion of independence covering families introduced by Lokshtanov et al. [25], which we believe could be of independent interest for the reconfiguration of independent sets. Let us start by formally defining such families and the various algorithms for extracting them on different graph classes.

39:4
On Finding Short Reconfiguration Sequences Between Independent Sets ▶ Theorem 1.4 ([25]). Let G be a graph such that G ∈ G, where G is a class of nowhere dense graphs. There is a deterministic algorithm that given k ≥ 1, runs in time O(f G (k) · (n + m) log n), and outputs an independence covering family for (G, k) of size at most O(g G (k) · n log n), where f G (k) and g G (k) depend on k and the class G but are independent of the size of the graph.
We use Theorems 1.2 and 1.3 to design fixed-parameter tractable algorithms for parameters k + ℓ + d and |N | + k + ℓ + d, respectively. Our algorithm for parameter |M | + ℓ + ∆ is based on the random separation technique [9]. Finally, we show that using independence covering families we can obtain a simpler and unified algorithm for the standard Token Jumping Reachability problem (a.k.a. Token Jumping) parameterized by k on both degenerate and nowhere dense classes of graphs; this is in contrast to the algorithms presented in [24]. To do so, we make use of Theorems 1.2 and 1.4. Note that the major difference between Theorems 1.2 and 1.4 is that in the former we are guaranteed a family of size at most O((kd) O(k) · log n) while in the latter the family is of size at least O(g G (k) · n log n), i.e., we have an extra linear dependence on n. This difference is the reason why our algorithm for parameter k + ℓ + d cannot be adapted to work for nowhere dense graphs. The current complexity status of all problems considered in this work is summarized in Table 1.

Preliminaries
Sets and functions. We denote the set of natural numbers (including 0) by N. For n ∈ N, we use [n] and [n] 0 to denote the sets {1, 2, · · · , n} and {0, 1, 2, · · · , n}, respectively. For a set X, we denote its power set by 2 An observation that we will make use of is the following: Graphs. Unless otherwise stated, we assume that each graph G is a simple, undirected graph with vertex set V (G) and edge set

FPT algorithm for parameter k + ℓ + d
In this section we start by designing a fixed-parameter tractable algorithm for the TSO problem parameterized by k + ℓ + d on d-degenerate graphs. We then show how the algorithm can be adapted for TJO as well as for parameter |N | + k + ℓ + d on graphs having a modulator N whose deletion leaves a d-degenerate graph (assuming N is given as part of the input).
. Without loss of generality, we assume that both S and T belong to F(G, k); as otherwise we can simply add them. Note that if (G, S, T, k, ℓ) is a yes-instance then there exists a sequence ⟨I 0 , I 1 , ..., I ℓ ⟩ of independent sets such that for all j ∈ {0, . . . , ℓ − 1} the set I j is an independent set of size k in G, I 0 = S, I ℓ = T , and I j+1 is obtained from I j by a token slide. This implies that there exists a sequence ⟨J 0 , In what follows, we assume that we guessed a sequence ⟨J 0 , J 1 , ..., J ℓ ⟩ of elements of F(G, k) such that J 0 = S and J ℓ = T . Our goal now is to design an algorithm that either finds a reconfiguration sequence ⟨I 0 = S, We define a constraint as a pair (X, b) where X ⊆ V (G) and b is a positive integer, called the budget of X. We denote a set of constraints by . We say that the set of constraints C is satisfied if all pairs (X, b) ∈ C are satisfied. We denote a set of sets of constraints by C. We now proceed by building sets of sets of constraints C 0 , C 1 , . . ., C ℓ and show that for each i ∈ [ℓ] 0 , the following invariants are satisfied: The total number of constraints at the i th step is C∈Ci |C| ≤ (i + 1)!.
At the base case, we let C 0 = {{(S, k)}}. The correctness of the base case immediately follows from its construction. We now proceed recursively as follows. Consider i ∈ [ℓ]. We assume that for each p ∈ [i − 1], we have computed C p that satisfy the correctness and size invariants.

I S A
We have c 0 = 1 from the base case of the algorithm. At each step the number of constraints in a set of constraints added increases by at most 1. For i = 0, we have only one constraint in {(S, k)} ∈ C 0 . Therefore, at the i th step, the maximum number of constraints in any set contained in C i is at most i + 1. In the i th recursive step of the algorithm, we add a new set of constraints C ′ corresponding to each constraint (X, b) contained in some member of C i−1 . So, we get the following recursive relation: |C i | = c i−1 . Using the fact that all members of C i contain at most i + 1 constraints, we get that c i = C∈Ci |C| ≤ (i + 1)|C i | = (i + 1)c i−1 . Solving the recurrence, we get c i ≤ (i + 1)!. Therefore, c ℓ ≤ (ℓ + 1)!. Let It can be seen that |Z ′ | = k and Z can be obtained from Z ′ by sliding one token. Since Z satisfies the set of constraints C, we have: The way we construct Z ′ , it must satisfy the following conditions: It can be clearly seen that Z ′ satisfies the set of constraints where the first equality follows from the fact that all X ′ such that (X ′ , b ′ ) ∈ C ′ are pairwise disjoint by Lemma 3.1 and the last equality follows from Lemma 3.1. Therefore, Thus, Z ′ is a k-sized subset of J i−1 and satisfies at least one set of constraints in C i−1 . By the induction hypothesis, Z ′ is reachable from S. Now, since Z is reachable from Z ′ , Z is also reachable from S. We now assume that the lemma holds true for i − 1, and prove it for i. Let C ∈ C i−1 be the set of constraints that where the first equality follows from the fact that all X such that (X, b) ∈ C are pairwise disjoint by Lemma 3.1 and the last equality follows again from Lemma 3.1.
In the i th step of the reconfiguration sequence, we slide a token from I ′ i−1 to I ′ i , i.e. from some set X such that (X, b) ∈ C to its open neighbourhood. Consider the set of constraints C ′ ∈ C i obtained by splitting the constraint (X, b) in the i th recursive step of the algorithm. We will show that where the first equality is because I ′ i ⊆ J i and the second equality is because I ′ i is obtained from I ′ i−1 by sliding a token from X to its neighbourhood. We have as one token is moved from X. When b = 1, we get I ′ i ∩ X = ∅ and this budget constraint is not included in the i th recursive step of the algorithm. For all other ( as none of the tokens in any X ′ are moved in the i th step of the reconfiguration sequence. Therefore, I ′ i = Z satisfies all the constraints in C ′ , as needed. ◀ We are now ready to prove our first main theorem. . We then add S and T to F(G, k) (in case they do not already belong to F(G, k)). Next, we "guess" a (iterate over every) sequence ⟨J 0 , J 1 , ..., J ℓ ⟩ of elements of F(G, k) such that J 0 = S, J ℓ = T . Note that this guessing can be accomplished in time O(((kd) O(k) · log n) ℓ+1 ), which by Proposition 2.1 is still FPT-time. Finally, we compute C 0 , C 1 , . . ., C ℓ , which by Lemma 3.2 can also be done in FPT-time. To conclude, we simply need to check whether T satisfies at least one set of constraints in C ℓ . The correctness of the algorithm follows from Lemma 3.3 and 3.4. ◀ ▶ Theorem 3.6. Token Sliding Optimization is fixed-parameter tractable parameterized by |N | + k + ℓ + d on graphs having a modulator N whose deletion leaves a d-degenerate graph (assuming N is given as part of the input).
Proof. We proceed exactly as in the proof of Theorem 3.5 but we invoke Theorem 1.3 instead of Theorem 1.

◀
We conlude this section by showing how we can adapt the previous two results for the Token Jumping Optimization problem. To allow tokens to jump to arbitrary vertices of the graph we only need to slightly modify our construction of the sets C 1 , . . ., C ℓ to obtain the following: ▶ Theorem 3.7. Token Jumping Optimization is fixed-parameter tractable parameterized by k + ℓ + d where d denotes the degeneracy of the graph and fixed-parameter tractable parameterized by |N | + k + ℓ + d on graphs having a modulator N whose deletion leaves a d-degenerate graph (assuming N is given as part of the input).

FPT algorithm for parameter |M | + ℓ + ∆
In this section, we prove that TSO and TJO are fixed-parameter tractable parameterized by |M | + ℓ + ∆. Recall that an instance of either problem is denoted by (G, S, T, k, ℓ) where V (G) can be partitioned into H and M and every vertex in H has degree at most ∆ in G.

39:8 On Finding Short Reconfiguration Sequences Between Independent Sets
Our algorithm is randomized and based on a variant of the color-coding technique [1] that is particularly useful in designing parameterized algorithms on graphs of bounded degree. The technique, known in the literature as random separation [9], boils down to a simple, but fruitful observation that in some cases, if we randomly color the vertex set of a graph using two colors, the solution or vertices we are looking for are appropriately colored with high probability. In our case, we want to make sure that the set of vertices involved in token slides or jumps gets highlighted. We note that our algorithm is an adaptation of the algorithm of Mouawad et al. [26] and it can easily be derandomized using standard techniques [10]. We start with an instance (G = (H, M, E), S, T, k, ℓ) of TSO; the algorithm is identical for TJO. We color independently every vertex of H using one of two colors, say red and blue (denoted by R and B), with probability 1 2 . We let χ : H → {R, B} denote the resulting random coloring. Suppose that (G, S, T, k, ℓ) is a yes-instance, and let σ denote a reconfiguration sequence from S to T of length at most ℓ. We say a vertex v ∈ H is touched in σ whenever a token slides from a neighbor of v to v or from v to some neighbor of v. We let V (σ) denote the set of vertices touched by σ. We say that the coloring χ is successful if both of the following conditions hold: Every vertex in V (σ) ∩ H is colored red; and Every vertex in N H (V (σ) ∩ H) is colored blue.
Observe that V (σ)∩H and N H (V (σ)∩H) are disjoint. Therefore, the two aforementioned conditions are independent. Moreover, since the maximum degree of Consequently, the probability that χ is successful is at least: If there exists a red vertex v which is not in S ∩ T and v belongs to a red component C of G[H R ] such that |V (C)| > 2ℓ then we delete v from the graph. We adjust S, T , and k appropriately to obtain the new equivalent instance (G ′ , S ′ , T ′ , k ′ , ℓ). Note that in this new instance (assuming a successful coloring) no vertices are colored blue and (assuming a correct guess) all vertices of M ′ will be touched in a solution. In other words, G ′ can be partitioned into M ′ and H ′ where H ′ consists of (an unbounded number of) connected components each consisting of at most 2ℓ vertices. Note that when the number of connected components is constant then we are done since we can solve the problem via brute-force. In other words, we can simply enumerate all possible sequences of length at most ℓ and make sure that at least one of them is the required reconfiguration sequence from S ′ to T ′ . This brute-force testing can be accomplished in time 2 O(ℓ log ℓ) · n O(1) .
Let us now consider the general case when the number of components is not necessarily bounded. We say a component C of There are at most 2ℓ important components. Hence, we only need to bound the number of unimportant components. To that end, we partition the unimportant components of G ′ − M ′ into equivalence classes with respect to the relation ≃. For two graphs G 1 , G 2 and two sets X 1 ⊆ V (G), X 2 ⊆ V (G 2 ), we say that (G 1 , X 1 ) and (G 2 , X 2 ) are isomorphic if the graphs G 1 and G 2 are isomorphic where vertices of X 1 and X 2 are now assigned the same color. Formally, a c-colored graph G is a tuple (V, E, K) such that K = {K 1 , . . . , K c } is a collection of subsets of V (G) where each K i is called a color set. Two colored graphs Let C 1 and C 2 be two components in G ′ − M ′ and let N 1 and N 2 be their respective neighborhoods in M ′ . We say C 1 and C 2 are equivalent, i.e., ▶ Lemma 4.2. The total number of 2-colored graphs with at most 2ℓ vertices is at most 2 O(ℓ 2 ) , and therefore, the equivalence relation ≃ has at most 2 O(ℓ 2 ) equivalence classes.
Assume that some equivalence class contains more than 2ℓ unimportant components. We claim that retaining only 2ℓ of them is enough. To see why, it is enough to note that V (σ) intersects with at most 2ℓ of those components; they are all equivalent. Putting it all together, we know that we have at most 2 O(ℓ 2 ) equivalence classes, each with at most 2ℓ components, and each component is of size at most 2ℓ. Hence, we can guess the sequence from S ′ to T ′ in time 2 O(ℓ 3 log ℓ) · n O(1) (testing whether two graphs with 2ℓ vertices are isomorphic can be accomplished naively in time 2 ℓ log ℓ ).
We have proven that the probability that χ is successful is at least 4 −ℓ∆ . Hence, to obtain a Monte Carlo algorithm with false negatives, we repeat the above procedure 4 ℓ∆ times and obtain the following result:

Hardness of TSO parameterized by ℓ on 2-degenerate graphs
In the Multicolored Clique problem, we are given an input graph G whose vertices are colored with k colors and the goal is to find a clique containing one vertex from each color. We show that TSO parameterized by ℓ is W[1]-hard on 2-degenerate graphs via a reduction from Multicolored Clique, known to be W[1]-hard.

39:10 On Finding Short Reconfiguration Sequences Between Independent Sets
We construct an instance (G ′ , S, T, κ, ℓ = 8 k 2 + 2k) of TSO from an instance of Multicolored Clique denoted by (G, k, (V 1 , V 2 , . . . , V k )), where, w.l.o.g., we assume that there are no edges between two vertices of G of the same color.
Construction of G ′ . We subdivide all the edges in G. Let the vertex set of G be V . All the vertices corresponding to the edges in G are partitioned into k 2 sets of the form E ij , where i = {1, 2, . . . , k} and j = {1, 2, . . . , k} and i ̸ = j, such that E ij contains all the vertices corresponding to the edges in G having one incident vertex of color i and the other incident vertex of color j. Let the union of all the sets E ij be denoted by E.
We introduce two independent sets X and Y , each of size k 2 . Let us label the vertices in X from 1 to k 2 and the sets E ij from 1 to k 2 . We add edges between vertex with label b in X and all the vertices in the E ij with label b. Similarly, we label the vertices of Y and add edges from each vertex in Y to all the vertices in the E ij having the same label. Each of these edges is further subdivided three times. Let the vertices on the subdivided edges from X to E, which are neither adjacent to some vertex in X nor E be denoted by U 1 , and the vertices on the subdivided edges from Y to E, which are neither adjacent to some vertex in Y nor E be denoted by U 2 . We take U = U 1 ∪ U 2 .
We also add a vertex corresponding to each vertex in V and add an edge between the two. Let this set of vertices be Z. The induced subgraph of G ′ having V ∪ Z as its vertex set forms a perfect matching. Our initial independent set S = V ∪ X ∪ U and our target independent Proof. Let the solution to the Multicolored Clique instance be {v 1 , v 2 , . . . , v k } ⊆ V . Consider the following reconfiguration sequence from S to T : 1. Slide each token on v i to its matched neighbour in Z; for a total of k slides.

2.
Since the vertices {v 1 , v 2 , . . . , v k } form a clique in G, there are k 2 edges, each having distinct pair of colors on their incident vertices. So in G ′ , all the vertices corresponding to the edges of the clique lie in distinct partitions E ij . We slide all the tokens from X to Y using these k 2 vertices. Consider the path from a vertex v x ∈ X to a vertex v y ∈ Y , passing through one of these k 2 vertices, say v i where i ∈ [k]. This path contains a vertex u 1 ∈ U 1 and a vertex u 2 ∈ U 2 . Slide the token on u 2 to v y (2 slides), the token on u 1 to u 2 through v i (4 slides), and the token on v x to u 1 along this path (2 slides); for a total of 8 k 2 slides. 3. Finally we slide the tokens in Z back to V ; for a total of k slides. The length of the reconfiguration sequence is 8 k 2 + 2k. This completes the proof. ◀ ▶ Lemma 5.3. If there is a reconfiguration sequence of length at most ℓ from S to T in G ′ then (G, k, (V 1 , V 2 , . . . , V k )) is a yes-instance of Multicolored Clique.
The combination of Lemmas 5.2 and 5.3 give us the following:

FPT algorithm for Token Jumping Reachability parameterized by k
We propose a generalized scheme for solving Token Jumping Reachability parameterized by k on graphs having a small k-independence covering family, i.e., a family of size O(f (k) · poly(n)). Degenerate and nowhere dense graphs admit such independence covering families as shown in [25]. We remove all the sets in the covering family of size less than k. We find out if the independent sets S and T are a part of the independence covering family. If not, we add them to the family. Let the size of the resulting k-independence covering family F(G, k) be q. For denoting an independent set in the family, we will use capital letters like, X, Y (⊆ V (G)). We construct a graph G with q vertices corresponding to the q sets in the family. Consider two independent sets I and I ′ in F(G, k). We add an edge between the vertices i and i ′ in G if and only if |I ∩ I ′ | ≥ k − 1. Note that for any two k-sized independent sets J and J ′ we can find a trivial reconfiguration sequence from J to J ′ if both of them are contained in some I in F(G, k).
In the algorithm, we find out if the vertices i s and i t in G are in the same connected component. If yes, then we know that S is reachable from T from the construction of G. Otherwise no reconfiguration sequence from S to T exists. ▶ Lemma 7.1. If there exists a path from i s to i t in G then there is a reconfiguration sequence from S to T in G.
Proof. Let i s = i 0 , i 1 , i 2 , . . . , i ℓ = i t be the path from i s to i t . We start the reconfiguration sequence with S. For each pair of vertices i j and i j+1 in the path, we have |I j ∩ I j+1 | ≥ k − 1 according to the construction. Now, let X j ⊆ I j be a k-sized independent set in the reconfiguration sequence and Y j ⊆ I j ∩ I j+1 be a (k − 1)-sized set. Let u j be a vertex in X j . We can obtain a k-sized independent set Z j = Y j ∪ {u j } from X j by at most k − 1 token jumps. Next we jump the token on u j to a vertex in I j+1 \ Y j to obtain a k-sized independent set X j+1 ⊆ I j+1 . This gives us a reconfiguration sequence from S to T , as needed. Proof. Let I ′ 1 , I ′ 2 , . . . , I ′ ℓ−1 be the sets in the covering family such that I i ⊆ I ′ i for i ∈ [ℓ − 1]. Since |I i ∩ I i+1 | = k − 1, we have |I ′ i ∩ I ′ i+1 | ≥ k − 1. If I ′ i and I ′ i+1 are the same set, then they correspond to the same vertex in G. Otherwise, they are connected by an edge according to the construction of G. We start from the vertex i s and following the reconfiguration sequence, we reach i t . This gives us a walk from i s to i t and a walk contains a path, as needed.