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- W2948340448 abstract "The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $ X times (V(G)setminus X)$ matrix over the binary field whose $(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in $V(G)setminus X$ and $0$ otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. This allows us to deduce that for each real $alpha$, the list of induced-subgraph-minimal graphs having average cut-rank larger than (or at least) $alpha$ is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cut-rank at most (or smaller than) $alpha$ for each real $alphage0$. Finally, we describe explicitly all graphs of average cut-rank at most $3/2$ and determine up to $3/2$ all possible values that can be realized as the average cut-rank of some graph." @default.
- W2948340448 created "2019-06-14" @default.
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- W2948340448 date "2020-12-01" @default.
- W2948340448 modified "2023-10-14" @default.
- W2948340448 title "The average cut-rank of graphs" @default.
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- W2948340448 doi "https://doi.org/10.1016/j.ejc.2020.103183" @default.
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