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Thus, it is a forbidden graph for the strict unit distance graphs, [20] but not one of the six forbidden graphs for the non-strict unit distance graphs. Other examples of graphs that are non-strict unit distance graphs but not strict unit distance graphs include the graph formed by removing an outer edge from , and the six-vertex graph formed ...
An example of a graph that is not a unit disk graph is the star, with one central node connected to six leaves: if each of six unit disks touches a common unit disk, some two of the six disks must touch each other. Therefore, unit disk graphs cannot contain an induced , subgraph. [1]
It has 104 edges and 52 vertices and is currently the smallest known example of a 4-regular matchstick graph. [3] It is a rigid graph. [4] Every 4-regular matchstick graph contains at least 20 vertices. [5] Examples of 4-regular matchstick graphs are currently known for all number of vertices ≥ 52 except for 53, 55, 56, 58, 59, 61 and 62.
For example, the Petersen graph can be drawn with unit edges in , but not in : its dimension is therefore 2 (see the figure to the right). This concept was introduced in 1965 by Paul Erdős, Frank Harary and William Tutte. [2] It generalises the concept of unit distance graph to more than 2 dimensions.
If a connected graph is a Cartesian product, it can be factorized uniquely as a product of prime factors, graphs that cannot themselves be decomposed as products of graphs. [2] However, Imrich & Klavžar (2000) describe a disconnected graph that can be expressed in two different ways as a Cartesian product of prime graphs:
A graph of the vertices of a pentagon, realized as an intersection graph of disks in the plane. This is an example of a graph with sphericity 2, also known as a unit disk graph . In graph theory , the sphericity of a graph is a graph invariant defined to be the smallest dimension of Euclidean space required to realize the graph as an ...
Graphs are special examples of metric spaces with their intrinsic path metric. Trees. If a tree is a path, its metric dimension is one. ... for unit interval graphs, ...
This example shows that a graph may require very different dimensions to be represented as a unit distance graph and as a strict unit distance graph. [2] The minimum number of complete bipartite subgraphs needed to cover the edges of a crown graph (its bipartite dimension, or the size of a minimum biclique cover) is