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Graph Basics

Nov 5

Contents

(Top)
1 Graphs
2 Graph Terms
3 Graph Characteristics and Types

Graphs

Graphs are a mathematical model comprising a set of vertices and a set of edges.

$G = (V, E)$

where $V$ is a set of vertices and $E$ is a set of edges.

In class, someone asked a question about gaming and graphs/trees. I couldn't think of the term, but this is what I was trying to describe: Behavior Trees).

Applications

Graph Terms

Undirected: edges do not have a direction; $(A, B)$ is the same as $(B, A)$
Directed: edges point in a specific direction; from one vertex to another
Weighted: edges have an associated weight/cost
Adjacent: two vertices are adjacent if they are connected by an edge
Incident: an edge is incident to a vertex if it is connected to the vertex
Degree: the degree of a vertex is counted as its number of incident edges
In-degree: (for directed graphs) the number of edges pointing to a given vertex
Out-degree: (for directed graphs) the number of edges pointing away from a given vertex
Path: a sequence of vertices ( $v_1$ , $v_2$ , ..., $v_k$ ) where there exists an edge connected each adjacent vertex ( $(v_1, v_2)$ , $(v_2, v_i)$ , ..., $(v_j, v_k)$ ) and there are not repeated edges
Cycle: a path where the first and last vertex are the same
Acyclic: a graph that does not contain any cycles
Connected: (for undirected graphs) a graph in which all vertices are connected by a path
Strongly Connected: (for directed graphs) a graph in which all vertices are connected by a path
Weakly Connected: (for directed graphs) a graph in which all vertices are connected if edges are replaced with undirected edges
Tree: a directed, acyclic graph that is connected and has a special vertex known as the root
DAG: shorthand for a directed, acyclic graph
Complete: a graph with an edges between every pair of vertices

Graph Characteristics and Types

Types

formatted by Markdeep 1.18

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