TY - GEN

T1 - Reconfiguration of spanning trees with many or few leaves

AU - Bousquet, Nicolas

AU - Ito, Takehiro

AU - Kobayashi, Yusuke

AU - Mizuta, Haruka

AU - Ouvrard, Paul

AU - Suzuki, Akira

AU - Wasa, Kunihiro

N1 - Funding Information:
Funding Partially supported by JSPS and MEAE-MESRI under the Japan-France Integrated Action Program (SAKURA). Nicolas Bousquet: This work was supported by ANR project GrR (ANR-18-CE40-0032). Takehiro Ito: Partially supported by JSPS KAKENHI Grant Numbers JP18H04091 and JP19K11814, Japan. Yusuke Kobayashi: Supported by JSPS KAKENHI Grant Numbers JP17K19960, JP18H05291, and JP20K11692, Japan. Haruka Mizuta: Partially supported by JSPS KAKENHI Grant Number JP19J10042, Japan. Paul Ouvrard: This work was supported by ANR project GrR (ANR-18-CE40-0032).
Funding Information:
Akira Suzuki: Partially supported by JSPS KAKENHI Grant Numbers JP18H04091 and JP20K11666, Japan. Kunihiro Wasa: Partially supported by JST CREST Grant Numbers JPMJCR18K3 and JP-MJCR1401, and JSPS KAKENHI Grant Number JP19K20350, Japan.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - Let G be a graph and T1,T2 be two spanning trees of G. We say that T1 can be transformed into T2 via an edge flip if there exist two edges e ∈ T1 and f in T2 such that T2 = (T1 \e)∪f. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed in [11]. We investigate the problem of determining, given two spanning trees T1,T2 with an additional property Π, if there exists an edge flip transformation from T1 to T2 keeping property Π all along. First we show that determining if there exists a transformation from T1 to T2 such that all the trees of the sequence have at most k (for any fixed k ≥ 3) leaves is PSPACE-complete. We then prove that determining if there exists a transformation from T1 to T2 such that all the trees of the sequence have at least k leaves (where k is part of the input) is PSPACE-complete even restricted to split, bipartite or planar graphs. We complete this result by showing that the problem becomes polynomial for cographs, interval graphs and when k = n-2.

AB - Let G be a graph and T1,T2 be two spanning trees of G. We say that T1 can be transformed into T2 via an edge flip if there exist two edges e ∈ T1 and f in T2 such that T2 = (T1 \e)∪f. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed in [11]. We investigate the problem of determining, given two spanning trees T1,T2 with an additional property Π, if there exists an edge flip transformation from T1 to T2 keeping property Π all along. First we show that determining if there exists a transformation from T1 to T2 such that all the trees of the sequence have at most k (for any fixed k ≥ 3) leaves is PSPACE-complete. We then prove that determining if there exists a transformation from T1 to T2 such that all the trees of the sequence have at least k leaves (where k is part of the input) is PSPACE-complete even restricted to split, bipartite or planar graphs. We complete this result by showing that the problem becomes polynomial for cographs, interval graphs and when k = n-2.

KW - Combinatorial reconfiguration

KW - PSPACE

KW - Polynomial-time algorithms

KW - Spanning trees

UR - http://www.scopus.com/inward/record.url?scp=85092474207&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85092474207&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2020.24

DO - 10.4230/LIPIcs.ESA.2020.24

M3 - Conference contribution

AN - SCOPUS:85092474207

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 28th Annual European Symposium on Algorithms, ESA 2020

A2 - Grandoni, Fabrizio

A2 - Herman, Grzegorz

A2 - Sanders, Peter

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 28th Annual European Symposium on Algorithms, ESA 2020

Y2 - 7 September 2020 through 9 September 2020

ER -