The extremal genus embedding of graphs

TL;DR

This paper investigates how adding a degree-3 vertex affects the genus of wheel graphs, establishes bounds on maximum genus using independent sets, and proposes algorithms for embeddings of complete graphs.

Contribution

It introduces a new method to determine the genus change in wheel graphs and provides improved bounds and algorithms for maximum genus embeddings of complete graphs.

Findings

Genus of Wn+v is 0 if neighbors are in the same face boundary; otherwise 1.

Provides a lower bound on maximum genus using independent sets.

Offers an algorithm for the number of maximum genus embeddings of Km.

Abstract

Let Wn be a wheel graph with n spokes. How does the genus change if adding a degree-3 vertex v, which is not in V (Wn), to the graph Wn? In this paper, through the joint-tree model we obtain that the genus of Wn+v equals 0 if the three neighbors of v are in the same face boundary of P(Wn); otherwise, {\deg}(Wn + v) = 1, where P(Wn) is the unique planar embedding of Wn. In addition, via the independent set, we provide a lower bound on the maximum genus of graphs, which may be better than both the result of D. Li & Y. Liu and the result of Z. Ouyang etc: in Europ. J. Combinatorics. Furthermore, we obtain a relation between the independence number and the maximum genus of graphs, and provide an algorithm to obtain the lower bound on the number of the distinct maximum genus embedding of the complete graph Km, which, in some sense, improves the result of Y. Caro and S. Stahl respectively.