TY - JOUR

T1 - Forbidden pairs with a common graph generating almost the same sets

AU - Chiba, Shuya

AU - Furuyal, Michitaka

AU - Fujisawa, Jun

AU - Ikarashi, Hironobu

N1 - Funding Information:
Partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B) 26800083 Partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B) 24340021 and Grant-in-Aid for Young Scientists (B) 26800085 Partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B) 26800086
Publisher Copyright:
© 2017, Australian National University. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2017/4/13

Y1 - 2017/4/13

N2 - Let H be a family of connected graphs. A graph G is said to be H-free if G does not contain any members of H as an induced subgraph. Let F(H) be the family of connected H-free graphs. In this context, the members of H are called forbidden subgraphs. In this paper, we focus on two pairs of forbidden subgraphs containing a com- mon graph, and compare the classes of graphs satisfying each of the two forbidden subgraph conditions. Our main result is the following: Let H1;H2;H3be connected graphs of order at least three, and suppose that H1 is twin-less. If the symmetric difference of F(fH1;H2g) and F(fH1;H3g) is finite and the tuple (H1;H2;H3) is non-trivial in a sense, then H2and H3are obtained from the same vertex-transitive graph by successively replacing a vertex with a clique and joining the neighbors of the original vertex and the clique. Furthermore, we refine a result in [Combin. Probab. Comput. 22 (2013) 733-748] concerning forbidden pairs.

AB - Let H be a family of connected graphs. A graph G is said to be H-free if G does not contain any members of H as an induced subgraph. Let F(H) be the family of connected H-free graphs. In this context, the members of H are called forbidden subgraphs. In this paper, we focus on two pairs of forbidden subgraphs containing a com- mon graph, and compare the classes of graphs satisfying each of the two forbidden subgraph conditions. Our main result is the following: Let H1;H2;H3be connected graphs of order at least three, and suppose that H1 is twin-less. If the symmetric difference of F(fH1;H2g) and F(fH1;H3g) is finite and the tuple (H1;H2;H3) is non-trivial in a sense, then H2and H3are obtained from the same vertex-transitive graph by successively replacing a vertex with a clique and joining the neighbors of the original vertex and the clique. Furthermore, we refine a result in [Combin. Probab. Comput. 22 (2013) 733-748] concerning forbidden pairs.

KW - Forbidden subgraph

KW - Star-free graph

KW - Vertex-transitive graph

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U2 - 10.37236/6190

DO - 10.37236/6190

M3 - Article

AN - SCOPUS:85018526888

VL - 24

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 2

ER -