Graph minor theory

graph formed from another graph by deleting edges and vertices and contracting edges

mapping of a graph into a tree

connected, bridgeless cubic graph with chromatic index equal to 4

mathematical theorem

thumb|The Goldner–Harary graph, an example of a planar 3-tree. In graph theory, a '''k-tree' is an undirected graph formed by starting with a (k + 1)-vertex complete graph and then repeatedly adding vertices in such a way that each added vertex v has exactly k neighbors U such that, together, the k + 1 vertices formed by v and U'' form a clique.

In graph theory, the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from being a tree. The smallest treewidth is 1; the graphs with treewidth 1 are exactly the trees and the forests. An example of graphs with treewidth at most 2 are the series–parallel graphs. The maximal graphs with treewidth exactly are called -trees, and the graphs with treewidth at most are called partial -trees. Many other well-studied graph families also have bounded treewidth.

In graph theory, a path decomposition of a graph is, informally, a representation of as a "thickened" path graph, and the pathwidth of is a number that measures how much the path was thickened to form . More formally, a path-decomposition is a sequence of subsets of vertices of such that the endpoints of each edge appear in one of the subsets and such that each vertex appears in a contiguous subsequence of the subsets, and the pathwidth is one less than the size of the largest set in such a decomposition. Pathwidth is also known as interval thickness (one less than the maximum clique size

conjecture that all graphs requiring k or more colors contain a k-vertex complete minor

characterization theorem in graph theory of planar graphs

mathematical theorem

graph that can be made planar by the removal of a single vertex

family of 7 undirected graphs

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

describing a family of graphs by excluding certain (sub)graphs

embedding a graph in 3D space with no cycles interlinked

on linear-time algorithms for graph logic

graph property