The average cut-rank of graphs

Abstract

The cut-rank of a set X of vertices in a graph G is defined as the rank of the X×(V(G)∖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)∖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 α, the list of induced-subgraph-minimal graphs having average cut-rank larger than (or at least) α 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) α for each real α≥0. 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.