On Girth and the Parameterized Complexity of Token Sliding and Token Jumping

Abstract

In the Token Jumping problem we are given a graph G = (V, E) and two independent sets S and T of G, each of size k ≥ 1. The goal is to determine whether there exists a sequence of k-sized independent sets in G, S0, S1, . . . , S , such that for every i, |Si| = k, Si is an independent set, S = S0, S = T , and |Si∆Si+1| = 2. In other words, if we view each independent set as a collection of tokens placed on a subset of the vertices of G, then the problem asks for a sequence of independent sets which transforms S to T by individual token jumps which maintain the independence of the sets. This problem is known to be PSPACE-complete on very restricted graph classes, e.g., planar bounded degree graphs and graphs of bounded bandwidth. A closely related problem is the Token Sliding problem, where instead of allowing a token to jump to any vertex of the graph we instead require that a token slides along an edge of the graph. Token Sliding is also known to be PSPACE-complete on the aforementioned graph classes. We investigate the parameterized complexity of both problems on several graph classes, focusing on the effect of excluding certain cycles from the input graph. In particular, we show that both Token Sliding and Token Jumping are fixed-parameter tractable on C4-free bipartite graphs when parameterized by k. For Token Jumping, we in fact show that the problem admits a polynomial kernel on {C3, C4}-free graphs. In the case of Token Sliding, we also show that the problem admits a polynomial kernel on bipartite graphs of bounded degree. We believe both of these results to be of independent interest. We complement these positive results by showing that, for any constant p ≥ 4, both problems are W[1]-hard on {C4, . . . , Cp}-free graphs and Token Sliding remains W[1]-hard even on bipartite graphs. ACM Subject ClassificationMathematics of computing → Paths and connectivity problems; Mathematics of computing → Graph algorithms; Mathematics of computing → Combinatoric problems

Independent set reconfiguration on directed graphs

“…Token Sliding is also studied from the viewpoint of fixed-parameter tractability (FPT). When the parameter is the number of tokens, the problem is W [1]-hard both on C 4 -free graphs and bipartite graphs, while it becomes FPT on their intersection, C 4 -free bipartite graphs [2]. In this paper, we show that Directed Token Sliding is W [1]-hard on DAGs.…”

“…We can construct a reconfiguration sequence from I s to I t in G as follows. If a token on v 1 slides to v 2 along the arc (v 1 , v 2 ) in G ′ , we do not move the tokens on G. Otherwise, if a token in G ′ moved from u i to v j for some u, v ∈ V (G) and i, j ∈ [2], and then we move the token on u to v. Since two vertices u and v are adjacent in G if and only if u i and v j are adjacent in G ′und for 1 ≤ i, j ≤ 2, implying that the intermediate sets in G are independent sets.…”

A survey on the parameterized complexity of the independent set and (connected) dominating set reconfiguration problems

“…. , C p }-free graphs [2], for any p ≥ 4. Having established hardness, we now focus on the parameterized complexity of ISR-TJ on sparse classes of graphs.…”

“…Graphs of bounded degree. We first explain why ISR-TJ is fixed-parameter tractable for the easy case of graphs G of maximum degree ∆ [2]. Let I s , I t be the source and target independent sets.…”

“…A set of vertices D ⊆ V (G) is a dominating set if every vertex in V (G) is either in D or has a neighbor in D. A dominating set D is a connected dominating set if additionally the graph induced by D is connected. Deciding whether a given graph contains a (connected) dominating set of a given size k is known to be NP-complete [42] and W [2]-complete parameterized by (solution size) k [22]. On the positive side, a very fruitful line of research [6,31,53,58] has shown that all three problems, i.e., Independent Set, Dominating Set, and Connected Dominating Set parameterized by solution size k become fixed-parameter tractable when restricted to sparse graph classes, such as planar, bounded degree, bounded treewidth, nowhere dense, and biclique-free classes of graphs.…”