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Essentially Tight Kernels for (Weakly) Closed Graphs
We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number c and weak closure number γ (Fox et al. SIAM J Comput 49(2):448–464, 2020) in addition to the standard param...
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Published in: | Algorithmica 2023-06, Vol.85 (6), p.1706-1735 |
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Main Authors: | , , |
Format: | Article |
Language: | English |
Subjects: | |
Citations: | Items that this one cites |
Online Access: | Get full text |
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Summary: | We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number
c
and weak closure number
γ
(Fox et al. SIAM J Comput 49(2):448–464, 2020) in addition to the standard parameter solution size
k
. The weak closure number
γ
of a graph is upper-bounded by the minimum of its closure number
c
and its degeneracy
d
. For
Capacitated Vertex Cover
,
Connected Vertex Cover
, and
Induced Matching
we obtain the first kernels of size
k
O
(
γ
)
,
k
O
(
γ
)
, and
(
γ
k
)
O
(
γ
)
, respectively. This extends previous results on the kernelization of these problems on degenerate graphs. These kernels are essentially tight as these problems are unlikely to admit kernels of size
k
o
(
γ
)
by previous results on their kernelization complexity on degenerate graphs (Cygan et al. ACM Trans Algorithms 13(3):43:1–43:22, 2017). For
Capacitated Vertex Cover
, we show that even a kernel of size
k
o
(
c
)
is unlikely. In contrast, for
Connected Vertex Cover
, we obtain a kernel with
O
(
c
k
2
)
vertices. Moreover, we prove that searching for an induced subgraph of order at least
k
belonging to a hereditary graph class
G
admits a kernel of size
k
O
(
γ
)
when
G
contains all complete and all edgeless graphs. Finally, we provide lower bounds for the kernelization of
Independent Set
on graphs with constant closure number
c
and kernels for
Dominating Set
on weakly closed split graphs and weakly closed bipartite graphs. |
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ISSN: | 0178-4617 1432-0541 |
DOI: | 10.1007/s00453-022-01088-7 |