Computing the Largest Bond of a Graph
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. Given s,t in V(G), an st-bond of G is a bond whose removal disconnects s and t. Contrasting with the large number of studies related to maximum cuts, there are 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 largest st-bond of a graph. Although cuts and bonds are similar, we remark that computing the largest bond of a graph tends to be harder than computing its maximum cut. We show that Largest Bond remains NP-hard even for planar bipartite graphs, and it does not admit a constant-factor approximation algorithm, unless P = NP. We also show that Largest Bond and Largest st-Bond on graphs of clique-width w cannot be solved in time f(w) x n^{o(w)} unless the Exponential Time Hypothesis fails, but they can be solved in time f(w) x n^{O(w)}. In addition, we show that both problems are fixed-parameter tractable when parameterized by the size of the solution, but they do not admit polynomial kernels unless NP subseteq coNP/poly.
bond
cut
maximum cut
connected cut
FPT
treewidth
clique-width
Mathematics of computing~Graph theory
Theory of computation~Parameterized complexity and exact algorithms
12:1-12:15
Regular Paper
Supported by Grant 2015/11937-9, São Paulo Research Foundation (FAPESP) and by Grant E-26/203.272/2017, Rio de Janeiro Research Foundation (FAPERJ) and by Grant 308689/2017-8, 425340/2016-3, 313026/2017-3, 422829/2018-8, 303726/2017-2, National Council for Scientific and Technological Development (CNPq).
A full version of the paper is available at http://arxiv.org/abs/1910.01071.
We thank the organizers of WoPOCA 2017 for the opportunity to bring together some of the co-authors of this paper.
Gabriel L.
Duarte
Gabriel L. Duarte
Fluminense Federal University, Rio de Janeiro, Brazil
Daniel
Lokshtanov
Daniel Lokshtanov
University of California Santa Barbara, CA, USA
Lehilton L. C.
Pedrosa
Lehilton L. C. Pedrosa
University of Campinas, São Paulo, Brazil
https://orcid.org/0000-0003-1001-082X
Rafael C. S.
Schouery
Rafael C. S. Schouery
University of Campinas, São Paulo, Brazil
https://orcid.org/0000-0002-0472-4810
Uéverton S.
Souza
Uéverton S. Souza
Fluminense Federal University, Rio de Janeiro, Brazil
https://orcid.org/0000-0002-5320-9209
10.4230/LIPIcs.IPEC.2019.12
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