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.
| Identifer | oai:union.ndltd.org:LSU/oai:etd.lsu.edu:etd-04042016-220803 |
| Date | 22 April 2016 |
| Creators | D'souza, Kimberly Sevin |
| Contributors | Ding, Guoli, Oxley, James, Oporowski, Bogdan, Litherland, Richard, Delzell, Charles, Terrell, Milton |
| Publisher | LSU |
| Source Sets | Louisiana State University |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lsu.edu/docs/available/etd-04042016-220803/ |
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