Hereditary biclique-Helly graphs: Recognition and maximal biclique enumeration
A biclique is a set of vertices that induce a complete bipartite graph. A graph G is biclique-Helly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also biclique-Helly, then G is hereditary biclique-Helly. A graph is C4-dominated when every cycle...
| Autor principal: | |
|---|---|
| Publicado: |
2013
|
| Materias: | |
| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_14627264_v15_n1_p55_Eguia http://hdl.handle.net/20.500.12110/paper_14627264_v15_n1_p55_Eguia |
| Aporte de: |
| id |
paper:paper_14627264_v15_n1_p55_Eguia |
|---|---|
| record_format |
dspace |
| spelling |
paper:paper_14627264_v15_n1_p55_Eguia2023-06-08T16:16:17Z Hereditary biclique-Helly graphs: Recognition and maximal biclique enumeration Soulignac, Francisco Juan Domination problems Hereditary biclique-Helly graphs Maximal bicliques Triangle-free graphs Biclique-helly Bicliques Complete bipartite graphs Domination problem Helly properties Induced cycle Induced subgraphs Triangle-free graphs Graphic methods Graph theory A biclique is a set of vertices that induce a complete bipartite graph. A graph G is biclique-Helly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also biclique-Helly, then G is hereditary biclique-Helly. A graph is C4-dominated when every cycle of length 4 contains a vertex that is dominated by the vertex of the cycle that is not adjacent to it. In this paper we show that the class of hereditary biclique-Helly graphs is formed precisely by those C4-dominated graphs that contain no triangles and no induced cycles of length either 5 or 6. Using this characterization, we develop an algorithm for recognizing hereditary biclique-Helly graphs in O(n2 +αm) time and O(n+m) space. (Here n, m, and α = O(m1/2) are the number of vertices and edges, and the arboricity of the graph, respectively.) As a subprocedure, we show how to recognize those C4-dominated graphs that contain no triangles in O(αm) time and O(n + m) space. Finally, we show how to enumerate all the maximal bicliques of a C4-dominated graph with no triangles in O(n2 + αm) time and O(αm) space. © 2013 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France. Fil:Soulignac, F.J. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2013 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_14627264_v15_n1_p55_Eguia http://hdl.handle.net/20.500.12110/paper_14627264_v15_n1_p55_Eguia |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Domination problems Hereditary biclique-Helly graphs Maximal bicliques Triangle-free graphs Biclique-helly Bicliques Complete bipartite graphs Domination problem Helly properties Induced cycle Induced subgraphs Triangle-free graphs Graphic methods Graph theory |
| spellingShingle |
Domination problems Hereditary biclique-Helly graphs Maximal bicliques Triangle-free graphs Biclique-helly Bicliques Complete bipartite graphs Domination problem Helly properties Induced cycle Induced subgraphs Triangle-free graphs Graphic methods Graph theory Soulignac, Francisco Juan Hereditary biclique-Helly graphs: Recognition and maximal biclique enumeration |
| topic_facet |
Domination problems Hereditary biclique-Helly graphs Maximal bicliques Triangle-free graphs Biclique-helly Bicliques Complete bipartite graphs Domination problem Helly properties Induced cycle Induced subgraphs Triangle-free graphs Graphic methods Graph theory |
| description |
A biclique is a set of vertices that induce a complete bipartite graph. A graph G is biclique-Helly when its family of maximal bicliques satisfies the Helly property. If every induced subgraph of G is also biclique-Helly, then G is hereditary biclique-Helly. A graph is C4-dominated when every cycle of length 4 contains a vertex that is dominated by the vertex of the cycle that is not adjacent to it. In this paper we show that the class of hereditary biclique-Helly graphs is formed precisely by those C4-dominated graphs that contain no triangles and no induced cycles of length either 5 or 6. Using this characterization, we develop an algorithm for recognizing hereditary biclique-Helly graphs in O(n2 +αm) time and O(n+m) space. (Here n, m, and α = O(m1/2) are the number of vertices and edges, and the arboricity of the graph, respectively.) As a subprocedure, we show how to recognize those C4-dominated graphs that contain no triangles in O(αm) time and O(n + m) space. Finally, we show how to enumerate all the maximal bicliques of a C4-dominated graph with no triangles in O(n2 + αm) time and O(αm) space. © 2013 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France. |
| author |
Soulignac, Francisco Juan |
| author_facet |
Soulignac, Francisco Juan |
| author_sort |
Soulignac, Francisco Juan |
| title |
Hereditary biclique-Helly graphs: Recognition and maximal biclique enumeration |
| title_short |
Hereditary biclique-Helly graphs: Recognition and maximal biclique enumeration |
| title_full |
Hereditary biclique-Helly graphs: Recognition and maximal biclique enumeration |
| title_fullStr |
Hereditary biclique-Helly graphs: Recognition and maximal biclique enumeration |
| title_full_unstemmed |
Hereditary biclique-Helly graphs: Recognition and maximal biclique enumeration |
| title_sort |
hereditary biclique-helly graphs: recognition and maximal biclique enumeration |
| publishDate |
2013 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_14627264_v15_n1_p55_Eguia http://hdl.handle.net/20.500.12110/paper_14627264_v15_n1_p55_Eguia |
| work_keys_str_mv |
AT soulignacfranciscojuan hereditarybicliquehellygraphsrecognitionandmaximalbicliqueenumeration |
| _version_ |
1768546643085361152 |