# Wolfram Language & System 10.3 (2015)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# AcyclicGraphQ

yields True if the graph g is an acyclic graph and False otherwise.

## DetailsDetails

• A graph has a cycle if there is a path that starts and ends in the same vertex.

## Background & ContextBackground & Context

• AcyclicGraphQ checks if a graph is cycle-less. A graph cycle (more properly called a circuit when the cycle is identified using an explicit path with specific endpoints) is a subset of a graph's edge set that forms a connected path such that the first node of the path corresponds to the last. A graph with no cycles is known as an acyclic graph, while a graph containing one or more cycles is called a cyclic graph. AcyclicGraphQ returns True for an acyclic graph (ignoring any self-loops) and False otherwise.
• A simple graph containing no cycles of length three is called a triangle-free graph, and a simple graph containing no cycles of length four is called a square-free graph. Simple acyclic graphs are therefore triangle-free and square-free. They are also non-Hamiltonian (i.e. they contain no Hamiltonian cycles).
• A connected acyclic graph is known as a tree. All trees are therefore acyclic by definition, and TreeGraphQ (which is equivalent to the logical conjunction of AcyclicGraphQ and ConnectedGraphQ) can be used to check if a graph is a tree. Trees appear extensively in computer science and in particular in the implementation of many types of algorithms and data structures, including file and folder storage on disk.
• A not-necessarily-connected acyclic graph is known as a forest. A forest with directed edges is more commonly known as a directed acyclic graph or DAG. A DAG is therefore a graph for which both AcyclicGraphQ and DirectedGraphQ return True. DAGs are important in modeling many different kinds of information, e.g. electronic circuits, information flows, and events and tasks.
• FindCycle can be used to find one or more cycles in graphs that are not acyclic (and returns the empty list for graphs that are).

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

Test whether a graph is acyclic:

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Not all graphs are acyclic:

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