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Every maximal planar graph on more than 3 vertices is at least 3-connected. [6] If a maximal planar graph has v vertices with v > 2, then it has precisely 3v – 6 edges and 2v – 4 faces. Apollonian networks are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles.
In the mathematical field of graph theory, the 26-fullerene graph is a polyhedral graph with V = 26 vertices and E = 39 edges. Its planar embedding has three hexagonal faces (including the one shown as the external face of the illustration) and twelve pentagonal faces. As a planar graph with only pentagonal and hexagonal faces, meeting in three ...
A graph is k-vertex-connected, but not necessarily planar, if and only if it has a convex embedding into (k −1)-dimensional space in which an arbitrary k-tuple of vertices are placed at the vertices of a simplex and, for each remaining vertex v, the convex hull of the neighbors of v is full-dimensional with v in its interior.
K 1 through K 4 are all planar graphs. However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph K 5 plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K 5 nor the complete bipartite ...
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
By Euler's formula for planar graphs, G has 3n − 6 edges; equivalently, if one defines the deficiency of a vertex v in G to be 6 − deg(v), the sum of the deficiencies is 12. Since G has at least four vertices and all faces of G are triangles, it follows that every vertex in G has degree at least three.
, is a graph with six vertices and nine edges, often referred to as the utility graph in reference to the problem. [1] It has also been called the Thomsen graph after 19th-century chemist Julius Thomsen. It is a well-covered graph, the smallest triangle-free cubic graph, and the smallest non-planar minimally rigid graph.
A graph is bipartite if and only if it contains no cycles of odd length. Since a tree contains no cycles at all, it is bipartite. Every tree with only countably many vertices is a planar graph. Every connected graph G admits a spanning tree, which is a tree that contains every vertex of G and whose edges are edges of G.