# K-vertex-connected graph

﻿
K-vertex-connected graph

In graph theory, a graph "G" with vertex set "V(G)" is said to be "k"-vertex-connected (or "k"-connected) if $G setminus X$ is connected for all $X subseteq V\left(G\right)$ with $left| X ight| < k$. In plain English, a graph is "k"-connected if the graph remains connected when you delete fewer than "k" vertices from the graph. Or, equivalently (owing to Menger's theorem), a graph is "k"-connected if any two of its vertices can be joined by "k" independent paths Harv|Diestel|2005| p=55.

A 1-vertex-connected graph is connected, while a 2-vertex-connected graph is said to be biconnected.

If a graph "G" is "k"-vertex-connected, and "k" < |"V(G)"|, then $k le delta\left(G\right)$, where "δ(G)" is the minimum degree of any vertex $v in V\left(G\right)$. This fact is clear since deleting all neighbors of a vertex of minimum degree will disconnect that vertex from the rest of the graph.

The 1-skeleton of any "k"-dimensional convex polytope forms a $k$-vertex-connected graph (Balinski 1961). As a partial converse, Steinitz showed that any 3-vertex-connected planar graph forms the skeleton of a convex polyhedron.

References

*cite journal
author = Balinski, M. L.
title = On the graph structure of convex polyhedra in "n"-space
journal = Pacific Journal of Mathematics
volume = 11
issue = 2
year = 1961
pages = 431–434
url = http://www.projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1103037323

*.

* k-edge-connected graph
* Connectivity (graph theory)
* Menger's theorem
* Structural cohesion

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• K-edge-connected graph — In graph theory, a graph G with edge set E(G) is said to be k edge connected if G setminus X is connected for all X subseteq E(G) with left| X ight| < k. In plain English, a graph is k edge connected if the graph remains connected when you delete …   Wikipedia

• Graph (mathematics) — This article is about sets of vertices connected by edges. For graphs of mathematical functions, see Graph of a function. For statistical graphs, see Chart. Further information: Graph theory A drawing of a labeled graph on 6 vertices and 7 edges …   Wikipedia

• Vertex (graph theory) — For other uses, see Vertex (disambiguation). A graph with 6 vertices and 7 edges where the vertex number 6 on the far left is a leaf vertex or a pendant vertex In graph theory, a vertex (plural vertices) or node is the fundamental unit out of… …   Wikipedia

• Graph toughness — In graph theory, toughness is a measure of the connectivity of a graph. A graph G is said to be t tough if, for every k > 1, G cannot be split into k different connected components by the removal of fewer than tk vertices. For instance, a graph… …   Wikipedia

• graph theory — Math. the branch of mathematics dealing with linear graphs. [1965 70] * * * Mathematical theory of networks. A graph consists of nodes (also called points or vertices) and edges (lines) connecting certain pairs of nodes. An edge that connects a… …   Universalium

• Graph of groups — In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of injective homomorphisms of the edge groups into the vertex groups.There is a… …   Wikipedia

• Connected component (graph theory) — A graph with three connected components. In graph theory, a connected component of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices. For example,… …   Wikipedia

• Graph theory — In mathematics and computer science, graph theory is the study of graphs : mathematical structures used to model pairwise relations between objects from a certain collection. A graph in this context refers to a collection of vertices or nodes and …   Wikipedia

• Connected dominating set — In graph theory, a connected dominated set and a maximum leaf spanning tree are two closely related structures defined on an undirected graph. Contents 1 Definitions 2 Complementarity 3 Algorithms 4 Applic …   Wikipedia

• Graph coloring — A proper vertex coloring of the Petersen graph with 3 colors, the minimum number possible. In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called colors to elements of a graph… …   Wikipedia