Xing, Yun2026-04-242026-04-242026-04-242026-04-22https://hdl.handle.net/10012/23052In this thesis, we study several problems related to excluded structures in matroids and tournaments. For matroids, we focus on finding the excluded minors for pinch-graphic matroids. For tournaments, we bound the size of excluded induced subgraphs of k-degenerate tournaments and we bound the number of backedge graphs of a tournament that are P3-free or Kt-free. Furthermore, we introduce a new way to color the arcs of a tournament and we propose a conjecture that connects this coloring to tournaments that do not contain Paley tournaments as an induced subgraph. For pinch-graphic matroids, we prove that if M is minimally non-pinch-graphic and M is not 3-connected, then |M| ≤ 20. Furthermore, we characterize all minimally non-pinch-graphic matroids that are not 3-connected. For the 3-connected case, we prove that in a special case, a 3-connected but not internally 4-connected minimally non-pinch-graphic matroid has at most 21 elements. We also improve a characterization of 3-separations in pinch-graphic matroids by Guenin and Heo. For tournaments, we show that an excluded induced subgraph for 1-in(out)-degenerate tournaments has at most (k² + 5k + 6) / 2 vertices, and an excluded induced subgraph for 1-degenerate tournaments has at most k² + 5k + 6 vertices. We also prove that a tournament with n vertices can have at most exponentially many P3-free or Kt-free backedge graphs, despite there being n! backedge graphs for a tournament with n vertices. We then introduce a new way to arc-color a tournament T with the corresponding chromatic number denoted as χ_Δ(T). The rule is to arc-color a tournament such that every directed cycle of length 3 gets 3 distinct colors. We prove that for tournaments T with n vertices, the growth rate of χ_Δ(T) is Θ(log n).enGraph TheoryMatroid TheoryTournamentsPinch-graphic MatroidsExcluded Induced SubgraphsExcluded MinorsMaster Thesis