### Prime vertex-minors of a prime graph [article]

Donggyu Kim, Sang-il Oum
2022 arXiv   pre-print
A split of a graph is a partition (A,B) of its vertex set such that min{|A|,|B|}≥ 2 and for some A'⊆ A and B'⊆ B, two vertices x∈ A and y∈ B are adjacent if and only if x∈ A' and y∈ B'. A graph is prime if it has no split. A vertex v of a graph is non-essential if at least two of the three kinds of vertex-minor reductions at v result in prime graphs. Allys (1994) proved that every prime graph with at least 5 vertices has a non-essential vertex unless it is locally equivalent to a cycle graph.
more » ... prove that every prime graph with at least 5 vertices has at least 2 non-essential vertices unless it is locally equivalent to a cycle graph. Furthermore, we show that every prime graph with at least 5 vertices has at least 3 non-essential vertices if and only if it is not locally equivalent to a graph with two specified vertices x and y consisting of at least two internally-disjoint paths from x to y in which x and y have no common neighbor.