graph theory, a connected graph is k-edge-connected if it remains connected whenever fewer than k edges are removed. The edge-connectivity of a graph...

In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer...

smallest edge cut disconnecting u from v. Again, local edge-connectivity is symmetric. A graph is called k-edge-connected if its edge connectivity is k or greater...

graph or k-edge-connected graph is a graph in which no set of k − 1 vertices (respectively, edges) exists that, when removed, disconnects the graph....

of graphs, systems of nodes or vertices connected in pairs by lines or edges. Contents:  A B C D E F G H I J K L M N O P Q R S T U V W X Y Z See also References...

words, a graph is edge-transitive if its automorphism group acts transitively on its edges. The number of connected simple edge-transitive graphs on n vertices...

the k-edge-connected Steiner network problem and the k-vertex-connected Steiner network problem, where the goal is to find a k-edge-connected graph or...

vertices can be any kind of object that is connected in pairs by edges. In the case of a directed graph, each edge has an orientation, from one vertex to...

In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected...

Similarly, an edge coloring assigns a color to each edges so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns...

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect...

of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of...

Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common...

A graph without cycles is called an acyclic graph. A directed graph without directed cycles is called a directed acyclic graph. A connected graph without...

field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete...

degree k is called a k‑regular graph or regular graph of degree k. Regular graphs of degree at most 2 are easy to classify: a 0-regular graph consists...

sample is a graph representation of a molecule, where atoms form the nodes and chemical bonds between atoms form the edges. In addition to the graph representation...

discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is...

specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs...

star with 3 edges is called a claw. The star Sk is edge-graceful when k is even and not when k is odd. It is an edge-transitive matchstick graph, and has...

In graph theory, the hypercube graph Qn is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q3...

In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes...

In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct...

coloring of the graph with two colors: if one colors all nodes in U {\displaystyle U} blue, and all nodes in V {\displaystyle V} red, each edge has endpoints...

discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair...

minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without...

graph is simple) connected in a closed chain. The cycle graph with n vertices is called Cn. The number of vertices in Cn equals the number of edges,...

{\displaystyle G} . The Cayley graph Γ = Γ ( G , S ) {\displaystyle \Gamma =\Gamma (G,S)} is an edge-colored directed graph constructed as follows: Each...

triangle-free graphs are bowtie-free graphs, since every butterfly contains a triangle. In a k-vertex-connected graph, an edge is said to be k-contractible...

In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph...