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- W3099641639 abstract "The cut-rank of a set X in a graph G is the rank of the 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⋅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." @default.
- W3099641639 created "2020-11-23" @default.
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- W3099641639 date "2023-02-01" @default.
- W3099641639 modified "2023-09-25" @default.
- W3099641639 title "Rank connectivity and pivot-minors of graphs" @default.
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- W3099641639 doi "https://doi.org/10.1016/j.ejc.2022.103634" @default.
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