Typer af grafer i datastruktur med eksempler

A graph is a non-linear data structure that consists of vertices and edges. The vertices contain the information or data, and the edges work as a link between a pair of vertices.

Graphs can be of multiple types, depending on the position of the nodes and edges. Here are some important types of Graphs:

Instrueret graf

The edges of the Directed Graph contain arrows that mean the direction. The arrow determines where the edge is pointed to or ends. Here is an example of the Directed Graph.

Instrueret graf

• Vi kan gå fra Node A til D.
• However, we cannot go from node D to node A, as the edge points from A to D.
• Da grafen ikke har vægte, vil det at rejse fra toppunkt A til D koste det samme som at rejse fra D til F.

Udirigeret graf

An Undirected Graph contains edges without pointers. It means we can travel vice versa between two vertices. Here is a simple example of the undirected Graph.

Udirigeret graf

I ovenstående graf,

• We can move from A to B.
• We can also move from B to A.
• Kanter indeholder ingen retninger.

It is an example of an undirected graph having a finite number of vertices and edges with no weights.

Vægtet graf

A Graph that contains weights or costs on the edges is called a weighted Graph. The numerical value generally represents the moving cost from one vertex to another vertex. Both Directed and Undirected Graphs can have weights on their edges. Here is an example of a weighted graph (Directed).

Instrueret graf med vægt

• A to B, there is an edge, and the weight is 5, which means moving from A to B will cost us 5.
• A points to B, but in this Graph, B has no direct edge over A. So, we cannot travel from B to A.
• However, if we want to move from A to F, there are multiple paths. The paths are ADF and ABF. ADF will cost (10+11) or 21.
• Here, the path ABF will cost (5+15) or 20. Here we are adding the weight of each edge in the path.

Here is an example of an Undirected Graph with weights:

Udirigeret graf med vægt

Her har kanten vægt, men ingen retning. Så det betyder at rejse fra toppunkt A til D vil koste 10 og omvendt.

Tovejs graf

Bi-directional and undirected graphs have a common property. That is:

• Generally, the undirected Graph can have one edge between two vertices.

For eksempel:

• Her vil det koste 10,- at flytte fra A til D eller D til A.
• I en tovejsgraf kan vi have to kanter mellem to hjørner.

Her er et eksempel:

Tovejs graf

Traveling from A to D will cost us 17, but traveling from D to A will cost us 12. So, we cannot assign two different weights if it is an undirected graph.

Uendelig graf

The Graph will contain an infinite number of edges and nodes. If a graph is Infinite and it is also a connected graph, then it will contain an infinite number of edges as well. Here, the extended edges mean that more edges might be connected to these nodes via edges. Here is an example of the infinite Graph:

Uendelig graf

Nul graf

A Null Graph contains only nodes or vertices but with no edges. If given a Graph G = (V, E), where V is vertices and E is edges, it will be null if the number of edges E is zero. Here is an example of a Null Graph:

Nul graf

Triviel graf

A graph data structure is considered trivial if only one vertex or node is present with no edges. Here is an example of a Trivial Graph:

Multi graf

A graph is called a multigraph when multiple edges are present between two vertices, or the vertex has a loop. The term “Loop” in Graph Data Structure means an edge pointing to the same node or vertex. A multigraph can be directed or undirected. Here is an example of a Multi Graph:

There are two edges from B to A. Moreover, vertex E has a self-loop. The above Graph is a directed graph with no weights on edges.

Komplet graf

A graph is complete if each vertex has directed or undirected edges with all other vertices. Suppose there is a total of V number of vertices and each vertex has exactly V-1 edges. Then, this Graph will be called a Complete Graph. In this type of Graph, each vertex is connected to all other vertices via edges. Here is an example of a Complete Graph with five vertices:

You can see in the image that the total number of nodes is five, and all the nodes have exactly four edges.

Forbundet graf

A Graph is called a Connected graph if we start from a node or vertex and can travel to all the nodes from the starting node. For this, there should be at least one edge between each pair of nodes or vertices. Here is an example of a Connected Graph:

Here is some explanation of the above Connected Graph:

• Assuming there is no edge between C and F, we cannot travel from A to G. However, the edge C to F enables us to travel to any node from a given node.
• En komplet graf er en forbundet graf, fordi vi kan flytte fra en node til en hvilken som helst anden node i den givne graf.

Cyklisk graf

A graph is said to be cyclic if there are one or more cycles present in the Graph. Here is an example of a Cyclic Graph:

Here, vertices A, B, and C form a cycle. A graph can have multiple cycles inside it.

Instrueret acyklisk graf (DAG)

A Graph is called a Directed Acyclic Graph or DAG if there are no cycles inside a graph. DAG is important while doing the Topologisk sortering or finding the execution order. DAG is also important for creating scheduling systems or scanning dependency of resources, etc. However, the Graph above does not contain any cycle inside. Here is a simple example of a Directed Acyclic Graph (DAG):

Cyklus graf

A Cycle Graph is not the same as the cyclic Graph. In a Cycle Graph, each node will have exactly two edges connected, meaning each node will have exactly two degrees. Here is an example of a Cycle Graph:

Bipartitegraf

Disse slags Grafer are special kinds of Graph where vertices are assigned to two sets. A Bipartite Graph must follow the rule:

• The two sets of vertices should be distinct, which means all the vertices must be divided into two groups or sets.
• Same-set vertices should not form any edges.

Euler graf

A Graph data structure is considered an Euler Graph if all the vertices have an even-numbered degree. The term degree of vertices means the number of edges pointing to or pointing out from a particular vertex. Here is an example of a Euler graph:

All the vertices have even degrees. Vertices A, D, E, and H have two degrees. Here, node C has four degrees, which is even.

Hamilton graf

A Hamilton Graph is a Connected Graph, where you can visit all the vertices from a given vertex without revisiting the same node or using the same edge. This kind of Connected Graph is known as the “Hamilton Graph”. The path you visit to verify if the given Graph is a Hamilton Graph or not is known as the Hamiltonian Path. Here is a simple graph example of a Hamilton:

På dette billede kan vi besøge alle hjørnerne fra enhver knude i ovenstående graf. En af stierne kan være ADCHBE. It is also possible to find a Hamilton Cycle. A Hamilton Cycle starts and ends at the same vertex. So, the Hamilton Cycle will be A-D-C-H-B-E-A.

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