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

07/03/2020
by   Valentin Bartier, et al.
0

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, ⟨ S_0, S_1, …, S_ℓ⟩, such that for every i, |S_i| = k, S_i is an independent set, S = S_0, S_ℓ = T, and |S_i Δ S_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 complement these positive results by showing that, for any constant p ≥ 4, both problems are W[1]-hard on C_ℓ-free graphs, where 4 ≤ℓ≤ p, and Token Sliding remains W[1]-hard even on bipartite graphs.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/02/2022

Token sliding on graphs of girth five

In the Token Sliding problem we are given a graph G and two independent ...
research
04/12/2022

Galactic Token Sliding

Given a graph G and two independent sets I_s and I_t of size k, the inde...
research
04/22/2020

Some results on Vertex Separator Reconfiguration

We present the first results on the complexity of the reconfiguration of...
research
06/30/2021

Parameterized Complexities of Dominating and Independent Set Reconfiguration

We settle the parameterized complexities of several variants of independ...
research
10/06/2022

Sequentially Swapping Tokens: Further on Graph Classes

We study the following variant of the 15 puzzle. Given a graph and two t...
research
07/09/2017

The complexity of independent set reconfiguration on bipartite graphs

We settle the complexity of the Independent Set Reconfiguration problem ...
research
08/15/2022

On The Complexity of Distance-d Independent Set Reconfiguration

For a fixed positive integer d ≥ 2, a distance-d independent set (DdIS) ...

Please sign up or login with your details

Forgot password? Click here to reset