TY - GEN

T1 - Parameterized Complexity of Perfectly Matched Sets

AU - Agrawal, Akanksha

AU - Bhattacharjee, Sutanay

AU - Jana, Satyabrata

AU - Sahu, Abhishek

N1 - Publisher Copyright:
© Akanksha Agrawal, Sutanay Bhattacharjee, Satyabrata Jana, and Abhishek Sahu.

PY - 2022/12/1

Y1 - 2022/12/1

N2 - For an undirected graph G, a pair of vertex disjoint subsets pA, Bq is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (“|B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2Op?kq ¨ nOp1q, and ii) Kb,b-free graphs. We obtain a linear kernel for planar graphs and kOpdq-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.

AB - For an undirected graph G, a pair of vertex disjoint subsets pA, Bq is a pair of perfectly matched sets if each vertex in A (resp. B) has exactly one neighbor in B (resp. A). In the above, the size of the pair is |A| (“|B|). Given a graph G and a positive integer k, the Perfectly Matched Sets problem asks whether there exists a pair of perfectly matched sets of size at least k in G. This problem is known to be NP-hard on planar graphs and W[1]-hard on general graphs, when parameterized by k. However, little is known about the parameterized complexity of the problem in restricted graph classes. In this work, we study the problem parameterized by k, and design FPT algorithms for: i) apex-minor-free graphs running in time 2Op?kq ¨ nOp1q, and ii) Kb,b-free graphs. We obtain a linear kernel for planar graphs and kOpdq-sized kernel for d-degenerate graphs. It is known that the problem is W[1]-hard on chordal graphs, in fact on split graphs, parameterized by k. We complement this hardness result by designing a polynomial-time algorithm for interval graphs.

KW - Apex-minor-free graphs

KW - d-degenerate graphs

KW - Interval Graphs

KW - Parameterized Complexity

KW - Perfectly Matched Sets

KW - Planar graphs

UR - http://www.scopus.com/inward/record.url?scp=85144205995&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.IPEC.2022.2

DO - 10.4230/LIPIcs.IPEC.2022.2

M3 - Conference contribution

AN - SCOPUS:85144205995

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 17th International Symposium on Parameterized and Exact Computation, IPEC 2022

A2 - Dell, Holger

A2 - Nederlof, Jesper

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

T2 - 17th International Symposium on Parameterized and Exact Computation, IPEC 2022

Y2 - 7 September 2022 through 9 September 2022

ER -