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1 The Digraphs package
 1.1 Introduction

1 The Digraphs package

1.1 Introduction

This is the manual for the Digraphs package version 0.5. This package was developed at the University of St Andrews by:

The Digraphs package contains a variety of methods for efficiently creating and storing digraphs and computing information about them. Full explanations of all the functions contained in the package are provided below.

If the Grape package is available, it will be loaded automatically. Digraphs created with the Digraphs package can be converted to Grape graphs with Graph (3.2-3), and conversely Grape graphs can be converted to Digraphs objects with Digraph (3.1-5). Grape is not required for Digraphs to run.

The bliss tool [JK07] is included in this package. It is an open-source tool for computing automorphism groups and canonical forms of graphs, written by Tommi Junttila and Petteri Kaski. Several of the methods in the Digraphs package rely on bliss.

For the purposes of this package and its documentation, the following definitions apply:

A digraph \(E=(E^0,E^1,r,s)\), also known as a directed graph, consists of a set of vertices \(E^0\) and a set of edges \(E^1\) together with functions \(s, r: E^1 \to E^0\), called the source and range, respectively. The source and range of an edge is respectively the values of \(s, r\) at that edge. An edge is called a loop if its source and range are the same. A digraph is called a multidigraph if there exist two or more edges with the same source and the same range.

A directed path on a digraph is a sequence of alternating vertices and edges \((v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)\) such that each edge \(e_i\) has source \(v_i\) and range \(v_{i+1}\), and no vertex is repeated. A directed walk is defined similarly, but vertices may be repeated. A cycle is defined similarly to a directed path, except that \(v_1 = v_n\). The length of a directed path, walk, or cycle \((v_1, e_1, v_2, e_2, ..., e_{n-1}, v_n)\) is equal to \(n-1\), the number of edges it contains.

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