TY - JOUR

T1 - Computing the Largest Bond and the Maximum Connected Cut of a Graph

AU - Duarte, Gabriel L.

AU - Eto, Hiroshi

AU - Hanaka, Tesshu

AU - Kobayashi, Yasuaki

AU - Kobayashi, Yusuke

AU - Lokshtanov, Daniel

AU - Pedrosa, Lehilton L.C.

AU - Schouery, Rafael C.S.

AU - Souza, Uéverton S.

N1 - Funding Information:
This work is partially supported by JST CREST JPMJCR1401, and JSPS KAKENHI Grant numbers JP17H01788, JP16K16010, JP17K19960, and JP19K21537, and by São Paulo Research Foundation (FAPESP) Grant number 2015/11937-9, and Rio de Janeiro Research Foundation (FAPERJ) Grant number E-26/203.272/2017, and by National Council for Scientific and Technological Development (CNPq-Brazil) Grant numbers 308689/2017-8, 425340/2016-3, 313026/2017-3, 422829/2018-8, 303726/2017-2. The Japanese authors thank Akitoshi Kawamura and Yukiko Yamauchi for giving an opportunity to discuss in the Open Problem Seminar in Kyushu University, Japan. Preliminary versions of this paper appeared in [] and [].
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

PY - 2021/5

Y1 - 2021/5

N2 - The cut-set ∂(S) of a graph G= (V, E) is the set of edges that have one endpoint in S⊂ V and the other endpoint in V\ S, and whenever G[S] is connected, the cut [S, V\ S] of G is called a connected cut. A bond of a graph G is an inclusion-wise minimal disconnecting set of G, i.e., bonds are cut-sets that determine cuts [S, V\ S] of G such that G[S] and G[V\ S] are both connected. Contrasting with a large number of studies related to maximum cuts, there exist very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond, and the maximum connected cut of a graph. Although cuts and bonds are similar, we remark that computing the largest bond and the maximum connected cut of a graph tends to be harder than computing its maximum cut. We show that it does not exist a constant-factor approximation algorithm to compute the largest bond, unless P=NP. Also, we show that Largest Bond and Maximum Connected Cut are NP-hard even for planar bipartite graphs, whereas Maximum Cut is trivial on bipartite graphs and polynomial-time solvable on planar graphs. In addition, we show that Largest Bond and Maximum Connected Cut are NP-hard on split graphs, and restricted to graphs of clique-width w they can not be solved in time f(w) no(w) unless the Exponential Time Hypothesis fails, but they can be solved in time f(w) nO(w). Finally, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, the treewidth, and the twin-cover number.

AB - The cut-set ∂(S) of a graph G= (V, E) is the set of edges that have one endpoint in S⊂ V and the other endpoint in V\ S, and whenever G[S] is connected, the cut [S, V\ S] of G is called a connected cut. A bond of a graph G is an inclusion-wise minimal disconnecting set of G, i.e., bonds are cut-sets that determine cuts [S, V\ S] of G such that G[S] and G[V\ S] are both connected. Contrasting with a large number of studies related to maximum cuts, there exist very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond, and the maximum connected cut of a graph. Although cuts and bonds are similar, we remark that computing the largest bond and the maximum connected cut of a graph tends to be harder than computing its maximum cut. We show that it does not exist a constant-factor approximation algorithm to compute the largest bond, unless P=NP. Also, we show that Largest Bond and Maximum Connected Cut are NP-hard even for planar bipartite graphs, whereas Maximum Cut is trivial on bipartite graphs and polynomial-time solvable on planar graphs. In addition, we show that Largest Bond and Maximum Connected Cut are NP-hard on split graphs, and restricted to graphs of clique-width w they can not be solved in time f(w) no(w) unless the Exponential Time Hypothesis fails, but they can be solved in time f(w) nO(w). Finally, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, the treewidth, and the twin-cover number.

KW - Bond

KW - Clique-width

KW - Connected cut

KW - Cut

KW - FPT

KW - Maximum cut

KW - Treewidth

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U2 - 10.1007/s00453-020-00789-1

DO - 10.1007/s00453-020-00789-1

M3 - Article

AN - SCOPUS:85099220213

SN - 0178-4617

VL - 83

SP - 1421

EP - 1458

JO - Algorithmica

JF - Algorithmica

IS - 5

ER -