#### Date of Award

Fall 11-11-2018

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Computational Analysis and Modeling

#### First Advisor

Jinko Kanno

#### Abstract

A graph is outer-planar (OP) if it has a plane embedding in which all of the vertices lie on the boundary of the outer face. A graph is near outer-planar (NOP) if it is edgeless or has an edge whose deletion results in an outer-planar graph. An edge of a non outer-planar graph whose removal results in an outer-planar graph is a vulnerable edge. This dissertation focuses on near outer-planar (NOP) graphs. We describe the class of all such graphs in terms of a ﬁnite list of excluded graphs, in a manner similar to the well-known Kuratowski Theorem for planar graphs. The class of NOP graphs is not closed by the minor relation, and the list of minimal excluded NOP graphs is not ﬁnite by the topological minor relation. Instead, we use the domination relation to deﬁne minimal excluded near outer-planar graphs, or XNOP graphs. To complete the list of 58 XNOP graphs, we give a description of those members of this list that dominate W3 or W4, wheels with three and four spokes, respectively.

To do this, we introduce the concepts of skeletons, joints and limbs. We ﬁnd an inﬁnite list of possible skeletons of XNOP graphs, as well as a ﬁnite list of possible limbs. With the list of skeletons, we permute the edges of a skeleton with the ﬁnite list of limbs to ﬁnd the complete list of XNOP graphs. In this process, we also develop algorithms in SageMath to prove the list of full-K4 XNOP graphs and prove that the list of skeletons of XNOP graphs is ﬁnite.

#### Recommended Citation

Lueder genannt Luehr, Tanya Allen, "" (2018). *Dissertation*. 6.

https://digitalcommons.latech.edu/dissertations/6