TY - JOUR

T1 - Forbidden subgraphs generating a finite set

AU - Fujisawa, Jun

AU - Plummer, Michael D.

AU - Saito, Akira

N1 - Funding Information:
The third author was partially supported by Japan Society for the Promotion of Science , Grant-in-Aid for Scientific Research (C), 22500018 , 2011.
Funding Information:
The first author was supported by Japan Society for the Promotion of Science , Grant-in-Aid for Young Scientists (B), 22740068 and Keio Gijuku Academic Development Funds.

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

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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 -