#### Identifier

etd-04042016-220803

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

A 3-connected graph $G$ is called weakly 4-connected if min $(|E(G_1)|, |E(G_2)|) \leq 4$ holds for all 3-separations $(G_1,G_2)$ of $G$. A 3-connected graph $G$ is called quasi 4-connected if min $(|V(G_1)|, |V(G_2)|) \leq 4$. We first discuss how to decompose a 3-connected graph into quasi 4-connected components. We will establish a chain theorem which will allow us to easily generate the set of all quasi 4-connected graphs. Finally, we will apply these results to characterizing all graphs which do not contain the Pyramid as a minor, where the Pyramid is the weakly 4-connected graph obtained by performing a $\Delta Y$ transformation to the octahedron. This result can be used to show an interesting characterization of quasi 4-connected, outer-projective graphs.

#### Date

2016

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

D'souza, Kimberly Sevin, "Excluding a Weakly 4-connected Minor" (2016). *LSU Doctoral Dissertations*. 1368.

https://digitalcommons.lsu.edu/gradschool_dissertations/1368

#### Committee Chair

Ding, Guoli