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 ksized 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 PSPACEcomplete 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 PSPACEcomplete 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 fixedparameter tractable on C_4free 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 ≥ 4, both problems are W[1]hard on {C_4, ..., C_p}free graphs and Token Sliding remains W[1]hard even on bipartite graphs.
BibTeX  Entry
@InProceedings{bartier_et_al:LIPIcs:2020:13388,
author = {Valentin Bartier and Nicolas Bousquet and Cl{\'e}ment Dallard and Kyle Lomer and Amer E. Mouawad},
title = {{On Girth and the Parameterized Complexity of Token Sliding and Token Jumping}},
booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)},
pages = {44:144:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771733},
ISSN = {18688969},
year = {2020},
volume = {181},
editor = {Yixin Cao and SiuWing Cheng and Minming Li},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/13388},
URN = {urn:nbn:de:0030drops133886},
doi = {10.4230/LIPIcs.ISAAC.2020.44},
annote = {Keywords: Combinatorial reconfiguration, Independent Set, Token Jumping, Token Sliding, Parameterized Complexity}
}
Keywords: 

Combinatorial reconfiguration, Independent Set, Token Jumping, Token Sliding, Parameterized Complexity 
Collection: 

31st International Symposium on Algorithms and Computation (ISAAC 2020) 
Issue Date: 

2020 
Date of publication: 

04.12.2020 