Prime vertex-minors of a prime graph
Donggyu Kim, Sang-il Oum

TL;DR
This paper investigates the structure of prime graphs and their vertex-minors, establishing the existence of multiple non-essential vertices unless the graph is locally equivalent to a cycle, with implications for prime graph reductions.
Contribution
It extends previous results by proving the existence of at least two non-essential vertices in prime graphs, and provides new bounds and conditions related to vertex-minors and pivot-minors.
Findings
Every prime graph with at least four vertices has at least two non-essential vertices unless it is locally equivalent to a cycle.
For prime graphs with at least six vertices, there exists a non-adjacent vertex whose removal or pivot operation preserves primality.
Prime graphs have at least three non-essential vertices unless they are locally equivalent to a specific multi-path structure.
Abstract
A graph is prime if it does not admit a partition of its vertex set such that and the rank of the submatrix of its adjacency matrix is at most . A vertex of a graph is non-essential if at least two of the three kinds of vertex-minor reductions at result in prime graphs. In 1994, Allys proved that every prime graph with at least four vertices has a non-essential vertex unless it is locally equivalent to a cycle graph. We prove that every prime graph with at least four vertices has at least two non-essential vertices unless it is locally equivalent to a cycle graph. As a corollary, we show that for a prime graph with at least six vertices and a vertex , there is a vertex such that or is prime, unless is adjacent to all other vertices and is isomorphic to a particular…
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Graph theory and applications · Advanced Computing and Algorithms
