TY - JOUR

T1 - Finding a maximum minimal separator

T2 - Graph classes and fixed-parameter tractability

AU - Hanaka, Tesshu

AU - Kobayashi, Yasuaki

AU - Kobayashi, Yusuke

AU - Yagita, Tsuyoshi

N1 - Funding Information:
This work is partially supported by JSPS KAKENHI Grant Number JP19K21537, JP20K19742, JP20H00595, JP18H05291, JP20K11692, and JST CREST JPMJCR1402.
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/4/14

Y1 - 2021/4/14

N2 - We study the problem of finding a maximum cardinality minimal separator of a graph. This problem is known to be NP-hard even for bipartite graphs. In this paper, we strengthen this hardness by showing that for planar bipartite graphs, the problem remains NP-hard. Moreover, for co-bipartite graphs and for line graphs, the problem also remains NP-hard. On the positive side, we give an algorithm deciding whether an input graph has a minimal separator of size at least k that runs in time 2O(k)nO(1). We further show that there is no 2o(n)nO(1)-time algorithm unless the Exponential Time Hypothesis (ETH) fails. Finally, we discuss a lower bound for polynomial kernelizations of this problem.

AB - We study the problem of finding a maximum cardinality minimal separator of a graph. This problem is known to be NP-hard even for bipartite graphs. In this paper, we strengthen this hardness by showing that for planar bipartite graphs, the problem remains NP-hard. Moreover, for co-bipartite graphs and for line graphs, the problem also remains NP-hard. On the positive side, we give an algorithm deciding whether an input graph has a minimal separator of size at least k that runs in time 2O(k)nO(1). We further show that there is no 2o(n)nO(1)-time algorithm unless the Exponential Time Hypothesis (ETH) fails. Finally, we discuss a lower bound for polynomial kernelizations of this problem.

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U2 - 10.1016/j.tcs.2021.03.006

DO - 10.1016/j.tcs.2021.03.006

M3 - Article

AN - SCOPUS:85102615079

VL - 865

SP - 131

EP - 140

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -