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A fast deterministic detection of small pattern graphs in graphs without large cliques

Kowaluk, Mirosław and Lingas, Andrzej LU (2019) In Theoretical Computer Science 770. p.79-87
Abstract

We show that for several pattern graphs on four vertices (e.g., C4), their induced copies in host graphs with n vertices and no clique on k+1 vertices can be deterministically detected in O(n2.5719k0.3176+n2k2) time for k<n0.394 and O(n2.5k0.5+n2k2) time for k≥n0.394. The aforementioned pattern graphs have a pair of non-adjacent vertices whose neighborhoods are equal. By considering dual graphs, in the same asymptotic time, we can also detect four vertex pattern graphs, that have an adjacent pair of vertices with the same neighbors among the remaining vertices (e.g., K4), in host graphs with n vertices... (More)

We show that for several pattern graphs on four vertices (e.g., C4), their induced copies in host graphs with n vertices and no clique on k+1 vertices can be deterministically detected in O(n2.5719k0.3176+n2k2) time for k<n0.394 and O(n2.5k0.5+n2k2) time for k≥n0.394. The aforementioned pattern graphs have a pair of non-adjacent vertices whose neighborhoods are equal. By considering dual graphs, in the same asymptotic time, we can also detect four vertex pattern graphs, that have an adjacent pair of vertices with the same neighbors among the remaining vertices (e.g., K4), in host graphs with n vertices and no independent set on k+1 vertices. By using the concept of Ramsey numbers, we can extend our method for induced subgraph isomorphism to include larger pattern graphs having a set of independent vertices with the same neighborhood and n-vertex host graphs without cliques on k+1 vertices (as well as the pattern graphs and host graphs dual to the aforementioned ones, respectively).

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Induced subgraph isomorphism, Matrix multiplication, Time complexity, Witnesses for Boolean matrix product
in
Theoretical Computer Science
volume
770
pages
79 - 87
publisher
Elsevier
external identifiers
  • scopus:85055747367
ISSN
0304-3975
DOI
10.1016/j.tcs.2018.10.028
language
English
LU publication?
yes
id
195b23eb-2b59-4483-9a4b-cdd3cebade32
date added to LUP
2018-11-20 12:20:36
date last changed
2019-05-27 19:14:18
@article{195b23eb-2b59-4483-9a4b-cdd3cebade32,
  abstract     = {<p>We show that for several pattern graphs on four vertices (e.g., C<sub>4</sub>), their induced copies in host graphs with n vertices and no clique on k+1 vertices can be deterministically detected in O(n<sup>2.5719</sup>k<sup>0.3176</sup>+n<sup>2</sup>k<sup>2</sup>) time for k&lt;n<sup>0.394</sup> and O(n<sup>2.5</sup>k<sup>0.5</sup>+n<sup>2</sup>k<sup>2</sup>) time for k≥n<sup>0.394</sup>. The aforementioned pattern graphs have a pair of non-adjacent vertices whose neighborhoods are equal. By considering dual graphs, in the same asymptotic time, we can also detect four vertex pattern graphs, that have an adjacent pair of vertices with the same neighbors among the remaining vertices (e.g., K<sub>4</sub>), in host graphs with n vertices and no independent set on k+1 vertices. By using the concept of Ramsey numbers, we can extend our method for induced subgraph isomorphism to include larger pattern graphs having a set of independent vertices with the same neighborhood and n-vertex host graphs without cliques on k+1 vertices (as well as the pattern graphs and host graphs dual to the aforementioned ones, respectively).</p>},
  author       = {Kowaluk, Mirosław and Lingas, Andrzej},
  issn         = {0304-3975},
  keyword      = {Induced subgraph isomorphism,Matrix multiplication,Time complexity,Witnesses for Boolean matrix product},
  language     = {eng},
  pages        = {79--87},
  publisher    = {Elsevier},
  series       = {Theoretical Computer Science},
  title        = {A fast deterministic detection of small pattern graphs in graphs without large cliques},
  url          = {http://dx.doi.org/10.1016/j.tcs.2018.10.028},
  volume       = {770},
  year         = {2019},
}