On girth and the parameterized complexity of token sliding and token jumping

TL;DR

This work investigates the parameterized complexity of Token Jumping ($TJ$) and Token Sliding ($TS$) reconfiguration problems for independent sets, focusing on graphs with forbidden cycles. It proves that both problems are fixed-parameter tractable on $C_4$-free bipartite graphs when parameterized by the number of tokens $k$, and it delivers polynomial kernels for several restricted graph classes: a quadratic kernel for $\\{C_3,C_4\\}$-free graphs in $TJ$, and a polynomial kernel (including $O(k^{12})$-vertex bounds) for $TS$ on bipartite $C_4$-free graphs, with a bounded-degree kernel as a stepping stone. Conversely, it establishes W[1]-hardness for both problems on $\\{C_4,\\dots, C_p\\}$-free graphs for any fixed $p\\ge 4$, and shows $TS$ is W[1]-hard on bipartite graphs, delineating the hardness introduced by cycle constraints. The results collectively illuminate how forbidding short cycles reshapes reconfiguration complexity and provide practical kernel bounds, contributing to the broader understanding of independent-set reconfiguration in sparse graph classes.

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 $k \geq 1$. The goal is to determine whether there exists a sequence of $k$-sized independent sets in $G$, $\langle S_0, S_1, \ldots, S_\ell \rangle$, such that for every $i$, $|S_i| = k$, $S_i$ is an independent set, $S = S_0$, $S_\ell = T$, and $|S_i ΔS_{i+1}| = 2$. 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 $C_4$-free bipartite graphs when parameterized by $k$. For Token Jumping, we in fact show that the problem admits a polynomial kernel on $\{C_3,C_4\}$-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 $p \geq 4$, both problems are W[1]-hard on $\{C_4, \dots, C_p\}$-free graphs and Token Sliding remains W[1]-hard even on bipartite graphs.

Figures (3)

• Figure 1: The construction of $G'$ for two cliques $V_i,V_j$ in $G$.
• Figure 2: Connections between the four copies of $V_p$ in $A \cup B$. Vertices with the same name are equivalent vertices. The red square represent tokens: two tokens are positioned on equivalent vertices at the left, and on non-equivalent vertices at the right.
• Figure 3: The constructed graph $G'$. Vertices in red are the vertices of $I_s$. An arrow between a vertex $v$ and a subset of vertices indicates that $v$ is complete to this subset. An arrow between a vertex $v$ and a brace indicates that $v$ is complete to the subsets included in the brace. A double arrow between two sets indicate these sets induce a complete bipartite graph. The connections between $A$ and $B_{end} \cup B_{start}$ are symmetric and have been omitted for the sake of clarity.

Theorems & Definitions (81)

• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 1
• proof
• Theorem 1
• proof
• Lemma 3: DBLP:conf/isaac/Fox-EpsteinHOU15
• Lemma 4: DBLP:conf/isaac/Fox-EpsteinHOU15