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Selected Problems On Matroid Minors, Jesse Taylor
Selected Problems On Matroid Minors, Jesse Taylor
LSU Doctoral Dissertations
This dissertation begins with an introduction to matroids and graphs. In the first chapter, we develop matroid and graph theory definitions and preliminary results sufficient to discuss the problems presented in the later chapters. These topics include duality, connectivity, matroid minors, and Cunningham and Edmonds's tree decomposition for connected matroids. One of the most well-known excluded-minor results in matroid theory is Tutte's characterization of binary matroids. The class of binary matroids is one of the most widely studied classes of matroids, and its members have many attractive qualities. This motivates the study of matroid classes that are close to being …
Combinatorial Minimal Free Resolutions Of Ideals With Monomial And Binomial Generators, Trevor Mcguire
Combinatorial Minimal Free Resolutions Of Ideals With Monomial And Binomial Generators, Trevor Mcguire
LSU Doctoral Dissertations
In recent years, the combinatorial properties of monomials ideals and binomial ideals have been widely studied. In particular, combinatorial interpretations of minimal free resolutions have been given in both cases. In this present work, we will generalize existing techniques to obtain two new results. If Lambda is an integer lattice in the n-dimensional integers satisfying some mild conditions, S is the polynomial ring with n variables and R is the group algebra of S[Lambda], then the first result is resolutions of Lambda-invariant submodules of the Laurent polynomial ring in n variables as R-modules. A consequence will be the ability to …
A Characterization Of Almost All Minimal Not Nearly Planar Graphs, Kwang Ju Choi
A Characterization Of Almost All Minimal Not Nearly Planar Graphs, Kwang Ju Choi
LSU Doctoral Dissertations
In this dissertation, we study nearly planar graphs, that is, graphs that are edgeless or have an edge whose deletion results in a planar graph. We show that all but finitely many graphs that are not nearly planar and do not contain one particular graph have a well-understood structure based on large Möbius ladders.