#### Date of Award

5-9-2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### First Advisor

Guantao Chen

#### Second Advisor

Hein van der Holst

#### Third Advisor

Yi Zhao

#### Fourth Advisor

Christian Avart

#### Abstract

In structural graph theory, connectivity is an important notation with a lot of applications. Tutte, in 1961, showed that a simple graph is 3-connected if and only if it can be generated from a wheel graph by repeatedly adding edges between nonadjacent vertices and applying vertex splitting. In 1971, Halin constructed a class of edge-minimal 3-connected planar graphs, which are a generalization of wheel graphs and later were named “Halin graphs” by Lovasz and Plummer. A Halin graph is obtained from a plane embedding of a tree with no stems having degree 2 by adding a cycle through its leaves in the natural order determined according to the embedding. Since Halin graphs were introduced, many useful properties, such as Hamiltonian, hamiltonian-connected and pancyclic, have been discovered. Hence, it will reveal many properties of a graph if we know the graph contains a spanning Halin subgraph. But unfortunately, until now, there is no positive result showing under which conditions a graph contains a spanning Halin subgraph. In this thesis, we characterize all forbidden pairs implying graphs containing spanning Halin subgraphs. Consequently, we provide a complete proof conjecture of Chen et al. Our proofs are based on Chudnovsky and Seymour’s decomposition theorem of claw-free graphs, which were published recently in a series of papers.

#### Recommended Citation

Yang, Ping, "Spanning Halin Subgraphs Involving Forbidden Subgraphs." Dissertation, Georgia State University, 2016.

http://scholarworks.gsu.edu/math_diss/29