Splitter Theorems for Graph Immersions

TL;DR

This paper develops splitter theorems for graph immersions in specific highly connected graph families, leading to new insights about the structure of graphs that contain certain complete bipartite graphs as immersions.

Contribution

It introduces splitter theorems for immersions in k-edge-connected and internally 4-edge-connected graphs, and proves a new immersion characterization involving K5 and K3,3.

Findings

Every 3-edge-connected, internally 4-edge-connected graph on at least seven vertices that immerses K5 also immerses K3,3.

Splitter theorems are established for k-edge-connected graphs with even k and for 3-edge-connected, internally 4-edge-connected graphs.

The results connect graph connectivity with the existence of specific graph immersions.

Abstract

We establish splitter theorems for graph immersions for two families of graphs, $k$-edge-connected graphs, with $k$ even, and 3-edge-connected, internally 4-edge-connected graphs. As a corollary, we prove that every $3$-edge-connected, internally $4$-edge-connected graph on at least seven vertices that immerses $K_5$ also has $K_{3,3}$ as an immersion.