Reductions of (v_3) configurations

TL;DR

This paper extends reduction methods for (v_3) configurations, showing that all such configurations can be reduced to either the Fano or Pappus configuration, simplifying their classification.

Contribution

It introduces a slightly extended set of reductions that unify the reduction process for all (v_3) configurations to two fundamental configurations.

Findings

All (v_3) configurations can be reduced to Fano or Pappus configurations.

Extended reductions simplify classification of (v_3) configurations.

Connection between combinatorial configurations and classical geometric configurations.

Abstract

Cubic bipartite graphs with girth at least 6 correspond to symmetric combinatorial (v_3) configurations. In 1887 V. Martinetti described a simple reduction method which enables one to reduce each combinatorial (v_3) configuration to one from the infinite set of so-called irreducible configurations. The aim of this paper is to show that a slightly extended set of reductions enables one to reduce each combinatorial (v_3) configuration either to the Fano configuration or to the Pappus configuration.