Vertex Splitting and Upper Embeddable Graphs

TL;DR

This paper establishes a new characterization of upper embeddability in graphs through vertex splitting operations, providing necessary and sufficient conditions, and offers an improved algorithm for determining upper embeddability.

Contribution

It introduces a novel relation between vertex splitting and upper embeddability, and constructs weak-minor-closed families of upper embeddable graphs, extending previous results.

Findings

Provides a necessary and sufficient condition for upper embeddability.

Develops a method to construct weak-minor-closed families from bouquets.

Reduces the complexity of algorithms for testing upper embeddability.

Abstract

The weak minor G of a graph G is the graph obtained from G by a sequence of edge-contraction operations on G. A weak-minor-closed family of upper embeddable graphs is a set G of upper embeddable graphs that for each graph G in G, every weak minor of G is also in G. Up to now, there are few results providing the necessary and sufficient conditions for characterizing upper embeddability of graphs. In this paper, we studied the relation between the vertex splitting operation and the upper embeddability of graphs; provided not only a necessary and sufficient condition for characterizing upper embeddability of graphs, but also a way to construct weak-minor-closed family of upper embeddable graphs from the bouquet of circles; extended a result in J: Graph Theory obtained by L. Nebesk{\P}y. In addition, the algorithm complex of determining the upper embeddability of a graph can be reduced much by the results obtained in this paper.