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 vertex cut. A graph is called k-vertex-connected or k-connected if its vertex connectivity is k or greater. More precisely, any graph G (complete...

vertex is called the apex. A k-apex graph is a graph that can be made planar by the removal of k vertices. 2.  Synonym for universal vertex, a vertex...

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...

specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set...

In graph theory, a vertex subset ⁠ S ⊂ V {\displaystyle S\subset V} ⁠ is a vertex separator (or vertex cut, separating set) for nonadjacent vertices a...

graph theory, a k-degenerate graph is an undirected graph in which every subgraph has at least one vertex of degree at most k: that is, some vertex in...

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...

the graph. A k-vertex-connected graph is often called simply a k-connected graph. A bipartite graph is a simple graph in which the vertex set can be partitioned...

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...

is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems...

In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular...

called an odd cycle. A cycle graph is: 2-edge colorable, if and only if it has an even number of vertices 2-regular 2-vertex colorable, if and only if it...

In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root. Both directed and...

and has odd vertex degrees, then L(G) is a vertex-transitive non-Cayley graph. If a graph G has an Euler cycle, that is, if G is connected and has an even...

the two-vertex complete graph, and may be decomposed into two copies of Qn – 1 connected to each other by a perfect matching. Hypercube graphs should not...

a graph G = (V, E), a vertex labeling is a function of V to a set of labels; a graph with such a function defined is called a vertex-labeled graph. Likewise...

object that is connected in pairs by edges. In the case of a directed graph, each edge has an orientation, from one vertex to another vertex. A path in a...

weighted directed graphs. The k-path partition problem is the problem of partitioning a given graph to a smallest collection of vertex-disjoint paths of...

and more 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...

Petersen graph. Only five connected vertex-transitive graphs with no Hamiltonian cycles are known: the complete graph K2, the Petersen graph, the Coxeter...

graph theory, an adjacent vertex of a vertex v in a graph is a vertex that is connected to v by an edge. The neighbourhood of a vertex v in a graph G...

entire graph G is this version: A graph is k-vertex-connected (it has more than k vertices and it remains connected after removing fewer than k vertices)...

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...

with one vertex and zero edges. More generally, a component of this type is formed for every isolated vertex in any graph. In a connected graph, there is...

mathematical 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...

In graph theory, a critical graph is an undirected graph all of whose proper subgraphs have smaller chromatic number. In such a graph, every vertex or...

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...

subdivision of a 3-vertex-connected planar graph. Tutte's spring theorem even states that for simple 3-vertex-connected planar graphs the position of the...

vertices in a k-vertex-connected graph there exists a cycle that passes through all the vertices in the set Gabriel Andrew Dirac (1925–1984), a graph theorist...