Independent Set Reconfiguration Research Articles

On the complexity of distance-d independent set reconfiguration

For a fixed positive integer d≥2, a distance-d independent set (DdIS) of a graph is a vertex-subset whose distance between any two members is at least d. Imagine that there is a token placed on each member of a DdIS. Two DdISs are adjacent under Token Sliding (TS) if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping (TJ), the target vertex needs not to be adjacent to the original one. The Distance-dIndependent Set Reconfiguration (DdISR) problem under TS/TJ asks if there is a corresponding sequence of adjacent DdISs that transforms one given DdIS into another. The problem for d=2, also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of DdISR on different graphs under TS and TJ for any fixed d≥3. On chordal graphs, we show that DdISR under TJ is in P when d is even and PSPACE-complete when d is odd. On split graphs, there is an interesting complexity dichotomy: DdISR is PSPACE-complete for d=2 but in P for d=3 under TS, while under TJ it is in P for d=2 but PSPACE-complete for d=3. Additionally, certain well-known hardness results for d=2 on perfect graphs and planar graphs of maximum degree three and bounded bandwidth can be extended for d≥3.

Extremal Independent Set Reconfiguration

The independent set reconfiguration problem asks whether one can transform one given independent set of a graph into another, by changing vertices one by one in such a way the intermediate sets remain independent. Extremal problems on independent sets are widely studied: for example, it is well known that an $n$-vertex graph has at most $3^{n/3}$ maximum independent sets (and this is tight). This paper investigates the asymptotic behavior of maximum possible length of a shortest reconfiguration sequence for independent sets of size $k$ among all $n$-vertex graphs. We give a tight bound for $k=2$. We also provide a subquadratic upper bound (using the hypergraph removal lemma) as well as an almost tight construction for $k=3$. We generalize our results for larger values of $k$ by proving an $n^{2\lfloor k/3 \rfloor}$ lower bound.

Galactic token sliding

Given a graph G and two independent sets Is and It of size k, the Independent Set Reconfiguration problem asks whether there exists a sequence of independent sets that transforms Is to It such that each independent set is obtained from the previous one using a so-called reconfiguration step. Viewing each independent set as a collection of k tokens placed on the vertices of a graph G, the two most studied reconfiguration steps are token jumping and token sliding. Over a series of papers, it was shown that the Token Jumping problem is fixed-parameter tractable (for parameter k) when restricted to sparse graph classes, such as planar, bounded treewidth, and nowhere dense graphs. As for the Token Sliding problem, almost nothing is known. We remedy this situation by showing that Token Sliding is fixed-parameter tractable on graphs of bounded degree, planar graphs, and chordal graphs of bounded clique number.

Distributed reconfiguration of maximal independent sets

We investigate a distributed maximal independent set reconfiguration problem, in which there are two MIS for which every node is given its membership status, and the nodes need to communicate with their neighbors to find a reconfiguration schedule from the first MIS to the second. We forbid two neighbors to change their membership status at the same step. We provide efficient solutions when the intermediate sets are only required to be independent and 4-dominating, which is almost always possible. Consequently, our goal is to pin down the tradeoff between the possible length of the schedule and the number of communication rounds. We prove that a constant length schedule can be found in O(MIS+R32) rounds. For bounded degree graphs, this is O(log⁎⁡n) rounds and we show that it is necessary. On the other extreme, we show that with a constant number of rounds we can find a linear length schedule.

Independent Set Reconfiguration Parameterized by Modular-Width

Independent Set Reconfiguration is one of the most well-studied problems in the setting of combinatorial reconfiguration. It is known that the problem is PSPACE-complete even for graphs of bounded bandwidth. This fact rules out the tractability of parameterizations by most well-studied structural parameters as most of them generalize bandwidth. In this paper, we study the parameterization by modular-width, which is not comparable with bandwidth. We show that the problem parameterized by modular-width is fixed-parameter tractable under all previously studied rules \({\mathsf {TAR}}\), \({\mathsf {TJ}}\), and \({\mathsf {TS}}\). The result under \({\mathsf {TAR}}\) resolves an open problem posed by Bonsma (J Graph Theory 83(2):164–195, 2016).

Token Sliding on Split Graphs

We consider the complexity of the Independent Set Reconfiguration problem under the Token Sliding rule. In this problem we are given two independent sets of a graph and are asked if we can transform one to the other by repeatedly exchanging a vertex that is currently in the set with one of its neighbors, while maintaining the set independent. Our main result is to show that this problem is PSPACE-complete on split graphs (and hence also on chordal graphs), thus resolving an open problem in this area. We then go on to consider the c-Colorable Reconfiguration problem under the same rule, where the constraint is now to maintain the set c-colorable at all times. As one may expect, a simple modification of our reduction shows that this more general problem is PSPACE-complete for all fixed c ≥ 1 on chordal graphs. Somewhat surprisingly, we show that the same cannot be said for split graphs: we give a polynomial time (nO(c)) algorithm for all fixed values of c, except c = 1, for which the problem is PSPACE-complete. We complement our algorithm with a lower bound showing that c-Colorable Reconfiguration is W[2]-hard on split graphs parameterized by c and the length of the solution, as well as a tight ETH-based lower bound for both parameters. Finally, we study c-Colorable Reconfiguration under a relaxed rule called Token Jumping, where exchanged vertices are not required to be adjacent. We show that the problem on chordal graphs is PSPACE-complete even if c is some fixed constant. We then show that the problem is polynomial-time solvable for strongly chordal graphs even if c is a part of the input.

Parameterized complexity of independent set reconfiguration problems

Suppose that we are given two independent sets I0 and Ir of a graph such that |I0|=|Ir|, and imagine that a token is placed on each vertex in I0. Then, the token jumping problem is to determine whether there exists a sequence of independent sets which transforms I0 into Ir so that each independent set in the sequence results from the previous one by moving exactly one token to another vertex. Therefore, all independent sets in the sequence must be of the same cardinality. This problem is PSPACE-complete even for planar graphs with maximum degree three. In this paper, we first show that the problem is W[1]-hard when parameterized only by the number of tokens. We then give an FPT algorithm for general graphs when parameterized by both the number of tokens and the maximum degree. Our FPT algorithm can be modified so that it finds an actual sequence of independent sets between I0 and Ir with the minimum number of token movements. We finally show that one of the results for token jumping can be extended to a more generalized reconfiguration problem for independent sets, called token addition and removal.

Reconfiguration of colorable sets in classes of perfect graphs

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (called a reconfiguration sequence) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration.

The Complexity of Independent Set Reconfiguration on Bipartite Graphs

We settle the complexity of the I ndependent S et R econfiguration problem on bipartite graphs under all three commonly studied reconfiguration models. We show that under the token jumping or token addition/removal model, the problem is NP-complete. For the token sliding model, we show that the problem remains PSPACE-complete.

Independent Set Reconfiguration in Cographs and their Generalizations

AbstractWe study the following independent set reconfiguration problem, called TAR‐Reachability: given two independent sets I and J of a graph G, both of size at least k, is it possible to transform I into J by adding and removing vertices one‐by‐one, while maintaining an independent set of size at least k throughout? This problem is known to be PSPACE‐hard in general. For the case that G is a cograph on n vertices, we show that it can be solved in time , and that the length of a shortest reconfiguration sequence from I to J is bounded by (if it exists). More generally, we show that if is a graph class for which (i) TAR‐Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in using disjoint union and complete join operations. Chordal graphs and claw‐free graphs are given as examples of such a class .