Joint-tree model and the maximum genus of graphs

TL;DR

This paper explores the properties of 1-critical-vertices in graphs using the joint-tree model, and investigates the maximum genus and upper embeddability of Spiral Snm graphs.

Contribution

It introduces a new approach using the joint-tree model to identify 1-critical-vertices and analyzes the maximum genus and embeddability of specific graph classes.

Findings

Identified types of 1-critical-vertices in graphs.

Established upper embeddability of Spiral Snm graphs.

Provided insights into the maximum genus related to 1-critical-vertices.

Abstract

The vertex v of a graph G is called a 1-critical-vertex for the maximum genus of the graph, or for simplicity called 1-critical-vertex, if G-v is a connected graph and {\deg}M(G - v) = {\deg}M(G) - 1. In this paper, through the joint-tree model, we obtained some types of 1-critical-vertex, and get the upper embeddability of the Spiral Snm .