Matroid theory

algorithm that makes locally optimal choices in a sequence of steps with the goal of reaching a global optimum

In combinatorics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite simple matroid is equivalent to a geometric lattice.

linearly independence of elements of a field extension that are also not related via finitary arithmetic operations

theorem that every finite set of points in the plane, not all collinear, has a line through exactly two points

finite projective plane of order 2

in linear algebra, the theorem that, for any set smaller than a spanning set, there is a set of vectors in the spanning set but missing from the smaller set that can be added to the smaller set to make that set spanning as well

the minimum number of edges to remove from a graph to eliminate all its cycles

field extension that contains an element algebraically independent from the base field

matroid whose independent sets are forests in an undirected graph

algebraic encoding of graph connectivity

thumb|upright=1.2|A 1-forest (a maximal pseudoforest), formed by three 1-trees In graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest.

partition of graph into sequence of paths

thumb|upright=1.35|Branch decomposition of a grid graph, showing an e-separation. The separation, the decomposition, and the graph all have width three. In graph theory, a branch-decomposition of an undirected graph G is a hierarchical clustering of the edges of G, represented by an unrooted binary tree T with the edges of G as its leaves. Removing any edge from T partitions the edges of G into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of any pair of subgraphs formed in this way. The branchwidth of G is the minimum width of any branch-decomposi