On switching nonseparable graphs with switching separable subgraphs

A graph of order $n \ge 4$ is called switching separable if its modulo-$2$ sum with some complete bipartite graph on the same set of vertices is divided into two mutually independent subgraphs, each having at least two vertices. We describe all switching nonseparable graphs of order $n$ whose induced subgraphs of order $(n-1)$ are all switching separable. In particular, such graphs exist only if $n$ is odd. This leads to the following essential refinement of the known test on switching separability, in terms of subgraphs: if all order-$(n-1)$ subgraphs of a graph of order $n$ are separable, then either the graph itself is separable, or $n$ is odd and the graph belongs to the two described switching classes.