Forbidden subgraphs generating a finite set

Jun Fujisawa, Michael D. Plummer, Akira Saito

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

For a set F of connected graphs, a graph G is said to be F -free if G does not contain any member of F as an induced subgraph. The members of F are referred to as forbidden subgraphs. When we study the relationship between forbidden subgraphs and a certain graph property, we often allow the possibility of the existence of exceptional graphs as long as their number is finite. However, in this type of research, if the set of k-connected F -free graphs itself, denoted by Gk(F ), is finite, then every graph in Gk(F ) logically satisfies all the graph properties, except for possibly a finite number of exceptions. If this occurs, F does not give any information about a particular property. We think that such sets F obscure the view in the study of forbidden subgraphs, and we want to remove them. With this motivation, we study the sets F with finite Gk(F ). We prove that if |F | ≤ 2 and Gk(F ) is finite, then either K1,2 F or F consists of a complete graph and a star. For each of the values of k, 1 ≤ k ≤ 6, we then characterize all the pairs {Kl, K1,m} such that Gk({Kl, K1,m}) is finite. We also give a complete characterization of F with |F | ≤ 3 and finite G2(F ).

Original languageEnglish
Pages (from-to)1835-1842
Number of pages8
JournalDiscrete Mathematics
Volume313
Issue number19
DOIs
Publication statusPublished - 2013

Fingerprint

Forbidden Subgraph
Stars
Finite Set
Graph in graph theory
Induced Subgraph
Complete Graph
Exception
Connected graph
Star

Keywords

  • Forbidden subgraphs
  • Hamiltonian cycles
  • K-connected graphs

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Forbidden subgraphs generating a finite set. / Fujisawa, Jun; Plummer, Michael D.; Saito, Akira.

In: Discrete Mathematics, Vol. 313, No. 19, 2013, p. 1835-1842.

Research output: Contribution to journalArticle

Fujisawa, Jun ; Plummer, Michael D. ; Saito, Akira. / Forbidden subgraphs generating a finite set. In: Discrete Mathematics. 2013 ; Vol. 313, No. 19. pp. 1835-1842.
@article{c21d4f3f3b534cb78cbb5f3debd917ed,
title = "Forbidden subgraphs generating a finite set",
abstract = "For a set F of connected graphs, a graph G is said to be F -free if G does not contain any member of F as an induced subgraph. The members of F are referred to as forbidden subgraphs. When we study the relationship between forbidden subgraphs and a certain graph property, we often allow the possibility of the existence of exceptional graphs as long as their number is finite. However, in this type of research, if the set of k-connected F -free graphs itself, denoted by Gk(F ), is finite, then every graph in Gk(F ) logically satisfies all the graph properties, except for possibly a finite number of exceptions. If this occurs, F does not give any information about a particular property. We think that such sets F obscure the view in the study of forbidden subgraphs, and we want to remove them. With this motivation, we study the sets F with finite Gk(F ). We prove that if |F | ≤ 2 and Gk(F ) is finite, then either K1,2 F or F consists of a complete graph and a star. For each of the values of k, 1 ≤ k ≤ 6, we then characterize all the pairs {Kl, K1,m} such that Gk({Kl, K1,m}) is finite. We also give a complete characterization of F with |F | ≤ 3 and finite G2(F ).",
keywords = "Forbidden subgraphs, Hamiltonian cycles, K-connected graphs",
author = "Jun Fujisawa and Plummer, {Michael D.} and Akira Saito",
year = "2013",
doi = "10.1016/j.disc.2012.05.015",
language = "English",
volume = "313",
pages = "1835--1842",
journal = "Discrete Mathematics",
issn = "0012-365X",
publisher = "Elsevier",
number = "19",

}

TY - JOUR

T1 - Forbidden subgraphs generating a finite set

AU - Fujisawa, Jun

AU - Plummer, Michael D.

AU - Saito, Akira

PY - 2013

Y1 - 2013

N2 - For a set F of connected graphs, a graph G is said to be F -free if G does not contain any member of F as an induced subgraph. The members of F are referred to as forbidden subgraphs. When we study the relationship between forbidden subgraphs and a certain graph property, we often allow the possibility of the existence of exceptional graphs as long as their number is finite. However, in this type of research, if the set of k-connected F -free graphs itself, denoted by Gk(F ), is finite, then every graph in Gk(F ) logically satisfies all the graph properties, except for possibly a finite number of exceptions. If this occurs, F does not give any information about a particular property. We think that such sets F obscure the view in the study of forbidden subgraphs, and we want to remove them. With this motivation, we study the sets F with finite Gk(F ). We prove that if |F | ≤ 2 and Gk(F ) is finite, then either K1,2 F or F consists of a complete graph and a star. For each of the values of k, 1 ≤ k ≤ 6, we then characterize all the pairs {Kl, K1,m} such that Gk({Kl, K1,m}) is finite. We also give a complete characterization of F with |F | ≤ 3 and finite G2(F ).

AB - For a set F of connected graphs, a graph G is said to be F -free if G does not contain any member of F as an induced subgraph. The members of F are referred to as forbidden subgraphs. When we study the relationship between forbidden subgraphs and a certain graph property, we often allow the possibility of the existence of exceptional graphs as long as their number is finite. However, in this type of research, if the set of k-connected F -free graphs itself, denoted by Gk(F ), is finite, then every graph in Gk(F ) logically satisfies all the graph properties, except for possibly a finite number of exceptions. If this occurs, F does not give any information about a particular property. We think that such sets F obscure the view in the study of forbidden subgraphs, and we want to remove them. With this motivation, we study the sets F with finite Gk(F ). We prove that if |F | ≤ 2 and Gk(F ) is finite, then either K1,2 F or F consists of a complete graph and a star. For each of the values of k, 1 ≤ k ≤ 6, we then characterize all the pairs {Kl, K1,m} such that Gk({Kl, K1,m}) is finite. We also give a complete characterization of F with |F | ≤ 3 and finite G2(F ).

KW - Forbidden subgraphs

KW - Hamiltonian cycles

KW - K-connected graphs

UR - http://www.scopus.com/inward/record.url?scp=84884817091&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884817091&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2012.05.015

DO - 10.1016/j.disc.2012.05.015

M3 - Article

AN - SCOPUS:84884817091

VL - 313

SP - 1835

EP - 1842

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 19

ER -