Non-Separating Cycles Through Prescirbed Vertices in Plane Graphs

dc.contributor.advisor Ye, Dong
dc.contributor.author Moser, Jacob
dc.contributor.committeemember Stephens, Chris
dc.contributor.committeemember Zha, Xiaoya
dc.date.accessioned 2024-04-24T22:02:31Z
dc.date.available 2024-04-24T22:02:31Z
dc.date.issued 2024
dc.date.updated 2024-04-24T22:02:31Z
dc.description.abstract A graph G is k-connected if it has at least k + 1 vertices and remains connected after deleting k − 1 vertices of G. A plane graph is a graph G with an embedding on a plane R2 such that the interior of an edge contains no vertex and no point of any other edge. When G is a plane graph, we call the regions of R2\G the faces of G. A planar graph is a graph isomorphic to a plane graph. A plane triangulation is a plane graph in which each face is bounded by a 3-cycle. A graph G is k-linked if, for any given 2k vertices there are k disjoint paths joining each pair of them. For a given graph H, a graph G is H-linked if, for every injective map from the vertices of H to the vertices of G, G contains a subdivision of H. Let (K4 − e) be the graph obtained from K4 by removing one edge. A graph G is (K4 − e)-linked if, for every injective map from the vertices of K4 − e to the vertices of G, G contains a subdivision of K4 − e. A graph is said to be k-cyclable if given any set of k vertices there is a cycle that contains the k vertices. We say that a graph is k-ordered or Ck-linked if given any set of k vertices there is a cycle through the k vertices in any specified order. A graph is (K2 ∪ K3)-linked if for every set of two vertices and every set of three vertices there exists a path joining the two vertices and a cycle on the three vertices. Seymour and Thomassen’s 2-linkage theorem characterizes all graphs which have two disjoint paths connecting any given two pair of vertices. Goddard proved that every 4-connected maximal planar graph is 4-ordered. Ellingham, Plummer, and Yu proved that any 4-connected planar triangulation is (K4 − e)-linked. In this thesis we completely characterize the obstructions to (K2 ∪ K3)-linkage in 4-connected plane triangulations. From these obstructions we can see that no 4-connected nor 5-connected plane triangulation is (K2 ∪ K3)-linked. iii
dc.description.degree M.S.
dc.identifier.uri https://jewlscholar.mtsu.edu/handle/mtsu/7184
dc.language.rfc3066 en
dc.publisher Middle Tennessee State University
dc.source.uri http://dissertations.umi.com/mtsu:11839
dc.subject 4-connected
dc.subject Cyclability
dc.subject Cyclable
dc.subject Linkage
dc.subject Planarity
dc.subject Triangulation
dc.subject Mathematics
dc.thesis.degreelevel masters
dc.title Non-Separating Cycles Through Prescirbed Vertices in Plane Graphs
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