Rank connectivity and pivot-minors of graphs

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The cut-rank of a set X in a graph G is the rank of the X x (V(G) - X) submatrix of the adjacency matrix over the binary field. A split is a partition of the vertex set into two sets (X, Y) such that the cut-rank of X is less than 2 and both X and Y have at least two vertices. A graph is prime (with respect to the split decomposition) if it is connected and has no splits. A graph G is k+l-rank-connected if for every set X of vertices with the cut-rank less than k, |X| or |V(G) - X| is less than k + l. We prove that every prime 3+2-rank-connected graph G with at least 10 vertices has a prime 3+3-rank-connected pivot-minor H such that |V(H)| = |V(G)| - 1. As a corollary, we show that every excluded pivot-minor for the class of graphs of rank-width at most k has at most (3.5 middot 6k - 1)/5 vertices for k >= 2. We also show that the excluded pivot-minors for the class of graphs of rank-width at most 2 have at most 16 vertices.(c) 2022 Elsevier Ltd. All rights reserved.
Publisher
ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
Issue Date
2023-02
Language
English
Article Type
Article
Citation

EUROPEAN JOURNAL OF COMBINATORICS, v.108

ISSN
0195-6698
DOI
10.1016/j.ejc.2022.103634
URI
http://hdl.handle.net/10203/304135
Appears in Collection
MA-Journal Papers(저널논문)
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