A fast deterministic detection of small pattern graphs in graphs without large cliques
(2019) In Theoretical Computer Science 770. p.7987 Abstract
We show that for several pattern graphs on four vertices (e.g., C_{4}), their induced copies in host graphs with n vertices and no clique on k+1 vertices can be deterministically detected in O(n^{2.5719}k^{0.3176}+n^{2}k^{2}) time for k<n^{0.394} and O(n^{2.5}k^{0.5}+n^{2}k^{2}) time for k≥n^{0.394}. The aforementioned pattern graphs have a pair of nonadjacent 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_{4}), in host graphs with n vertices... (More)
We show that for several pattern graphs on four vertices (e.g., C_{4}), their induced copies in host graphs with n vertices and no clique on k+1 vertices can be deterministically detected in O(n^{2.5719}k^{0.3176}+n^{2}k^{2}) time for k<n^{0.394} and O(n^{2.5}k^{0.5}+n^{2}k^{2}) time for k≥n^{0.394}. The aforementioned pattern graphs have a pair of nonadjacent 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_{4}), 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 nvertex host graphs without cliques on k+1 vertices (as well as the pattern graphs and host graphs dual to the aforementioned ones, respectively).
(Less)
 author
 Kowaluk, Mirosław and Lingas, Andrzej ^{LU}
 organization
 publishing date
 2019
 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
 03043975
 DOI
 10.1016/j.tcs.2018.10.028
 language
 English
 LU publication?
 yes
 id
 195b23eb2b5944839a4bcdd3cebade32
 date added to LUP
 20181120 12:20:36
 date last changed
 20190527 19:14:18
@article{195b23eb2b5944839a4bcdd3cebade32, 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<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 nonadjacent 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 nvertex 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 = {03043975}, keyword = {Induced subgraph isomorphism,Matrix multiplication,Time complexity,Witnesses for Boolean matrix product}, language = {eng}, pages = {7987}, 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}, }