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The complete list of all free trees on 2, 3, and 4 labeled vertices: = tree with 2 vertices, = trees with 3 vertices, and = trees with 4 vertices.. In combinatorics, an area of mathematics, graph enumeration describes a class of combinatorial enumeration problems in which one must count undirected or directed graphs of certain types, typically as a function of the number of vertices of the ...
A graph is called k-vertex-connected or k-connected if its vertex connectivity is k or greater. More precisely, any graph G (complete or not) is said to be k -vertex-connected if it contains at least k + 1 vertices, but does not contain a set of k − 1 vertices whose removal disconnects the graph; and κ ( G ) is defined as the largest k such ...
If a planar graph is 3-connected, it has a unique planar embedding up to the choice of which face is the outer face and of orientation of the embedding: the faces of the embedding are exactly the nonseparating cycles of the graph. However, for a planar graph (with labeled vertices and edges) that is 2-connected but not 3-connected, there may be ...
A graph with connectivity 4. 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 than k vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest k for which the graph is k-vertex-connected.
The "compulsory" edges of the fragments, that must be part of any Hamiltonian path through the fragment, are connected at the central vertex; because any cycle can use only two of these three edges, there can be no Hamiltonian cycle. The resulting Tutte graph is 3-connected and planar, so by Steinitz' theorem it is the graph of a polyhedron. In ...
The connectivity of a graph is the minimum number of vertices that must be removed to disconnect it. Equivalently, the connectivity of a graph is the greatest integer k for which the graph is k-connected. While terminology varies, noun forms of connectedness-related properties often include the term connectivity. Thus, when discussing simply ...
In 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 is the largest k for which the graph is k-edge-connected. Edge connectivity and the enumeration of k-edge-connected graphs was studied by Camille Jordan in 1869. [1]
The polyhedral graph formed as the Schlegel diagram of a regular dodecahedron. In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected, planar graphs.