Traversals and coloring

bfs_parents(g, s[; dir=:out])

Perform a breadth-first search of graph g starting from vertex s. Return a vector of parent vertices indexed by vertex. If dir is specified, use the corresponding edge direction (:in and :out are acceptable values).

Performance

This implementation is designed to perform well on large graphs. There are implementations which are marginally faster in practice for smaller graphs, but the performance improvements using this implementation on large graphs can be significant.

bfs_tree(g, s[; dir=:out])

Provide a breadth-first traversal of the graph g starting with source vertex s, and return a directed acyclic graph of vertices in the order they were discovered. If dir is specified, use the corresponding edge direction (:in and :out are acceptable values).

gdistances!(g, source, dists; sort_alg=QuickSort)

Fill dists with the geodesic distances of vertices in g from source vertex (or collection of vertices) source. dists should be a vector of length nv(g) filled with typemax(T). Return dists.

For vertices in disconnected components the default distance is typemax(T).

An optional sorting algorithm may be specified (see Performance section).

Performance

gdistances uses QuickSort internally for its default sorting algorithm, since it performs the best of the algorithms built into Julia Base. However, passing a RadixSort (available via SortingAlgorithms.jl) will provide significant performance improvements on larger graphs.

gdistances(g, source; sort_alg=QuickSort)

Return a vector filled with the geodesic distances of vertices in g from source. If source is a collection of vertices each element should be unique. For vertices in disconnected components the default distance is typemax(T).

An optional sorting algorithm may be specified (see Performance section).

Performance

gdistances uses QuickSort internally for its default sorting algorithm, since it performs the best of the algorithms built into Julia Base. However, passing a RadixSort (available via SortingAlgorithms.jl) will provide significant performance improvements on larger graphs.

has_path(g::AbstractGraph, u, v; exclude_vertices=Vector())

Return true if there is a path from u to v in g (while avoiding vertices in exclude_vertices) or u == v. Return false if there is no such path or if u or v is in excluded_vertices.

tree(parents)

Convert a parents array into a directed graph.

bipartite_map(g) -> Vector{UInt8}

For a bipartite graph g, return a vector c of size $|V|$ containing the assignment of each vertex to one of the two sets ($c_i == 1$ or $c_i == 2$). If g is not bipartite, return an empty vector.

Implementation Notes

Note that an empty vector does not necessarily indicate non-bipartiteness. An empty graph will return an empty vector but is bipartite.

Examples

julia> using Graphs

julia> g = SimpleGraph(3);

julia> bipartite_map(g)
3-element Vector{UInt8}:
 0x01
 0x01
 0x01

julia> add_vertices!(g, 3);

julia> add_edge!(g, 1, 2);

julia> add_edge!(g, 2, 3);

julia> bipartite_map(g)
6-element Vector{UInt8}:
 0x01
 0x02
 0x01
 0x01
 0x01
 0x01

is_bipartite(g)

Return true if graph g is bipartite.

Examples

julia> using Graphs

julia> g = SimpleGraph(3);

julia> add_edge!(g, 1, 2);

julia> add_edge!(g, 2, 3);

julia> is_bipartite(g)
true

julia> add_edge!(g, 1, 3);

julia> is_bipartite(g)
false

dfs_parents(g, s[; dir=:out])

Perform a depth-first search of graph g starting from vertex s. Return a vector of parent vertices indexed by vertex. If dir is specified, use the corresponding edge direction (:in and :out are acceptable values).

Implementation Notes

This version of DFS is iterative.

dfs_tree(g, s)

Return an ordered vector of vertices representing a directed acyclic graph based on depth-first traversal of the graph g starting with source vertex s.

is_cyclic(g)

Return true if graph g contains a cycle.

Implementation Notes

The algorithm uses a DFS. Self-loops are counted as cycles.

topological_sort(g)

Return a topological sort of a directed graph g as a vector of vertices in topological order.

Implementation Notes

This is currently just an alias for topological_sort_by_dfs

topological_sort_by_dfs(g)

Return a topological sort of a directed graph g as a vector of vertices in topological order.

diffusion(g, p, n)

Run diffusion simulation on g for n steps with spread probabilities based on p. Return a vector with the set of new vertices reached at each step of the simulation.

Optional Arguments

• initial_infections=sample(vertices(g), 1): A list of vertices that

are infected at the start of the simulation.

• watch=Vector(): While simulation is always run on the full graph,

specifying watch limits reporting to a specific set of vertices reached during the simulation. If left empty, all vertices will be watched.

• normalize=false: if false, set the probability of spread from a vertex $i$ to

each of the outneighbors of $i$ to $p$. If true, set the probability of spread from a vertex $i$ to each of the outneighbors of $i$ to $\frac{p}{outdegreee(g, i)}$.

• rng=nothing: A random generator to sample from.

diffusion_rate(results)
diffusion_rate(g, p, n; ...)

Given the results of a diffusion output or the parameters to the diffusion simulation itself, (run and) return the rate of diffusion as a vector representing the cumulative number of vertices infected at each simulation step, restricted to vertices included in watch, if specified.

struct Coloring{T}

Store the number of colors used and mapping from vertex to color

degree_greedy_color(g)

Color graph g iteratively in the descending order of the degree of the vertices.

greedy_color(g; sort_degree=false, reps = 1)

Color graph g based on Greedy Coloring Heuristics

The heuristics can be described as choosing a permutation of the vertices and assigning the lowest color index available iteratively in that order.

If sort_degree is true then the permutation is chosen in reverse sorted order of the degree of the vertices. parallel and reps are irrelevant in this case.

If sort_degree is false then reps colorings are obtained based on random permutations and the one using least colors is chosen.

perm_greedy_color(g, seq)

Color graph g according to an order specified by seq using a greedy heuristic. seq[i] = v implies that vertex v is the $i^{th}$ vertex to be colored.

random_greedy_color(g, reps)

Color the graph g iteratively in a random order using a greedy heuristic and choose the best coloring out of reps such random colorings.

maximum_adjacency_visit(g[, distmx][, log][, io][, s])
maximum_adjacency_visit(g[, s])

Return the vertices in g traversed by maximum adjacency search, optionally starting from vertex s (default 1). An optional distmx matrix may be specified; if omitted, edge distances are assumed to be 1. If log (default false) is true, visitor events will be printed to io, which defaults to STDOUT; otherwise, no event information will be displayed.

mincut(g, distmx=weights(g))

Return a tuple (parity, bestcut), where parity is a vector of integer values that determines the partition in g (1 or 2) and bestcut is the weight of the cut that makes this partition. An optional distmx matrix of non-negative weights may be specified; if omitted, edge distances are assumed to be 1.

non_backtracking_randomwalk(g, s, niter; rng=nothing, seed=nothing)

Perform a non-backtracking random walk on directed graph g starting at vertex s and continuing for a maximum of niter steps. Return a vector of vertices visited in order.

randomwalk(g, s, niter; rng=nothing, seed=nothing)

Perform a random walk on graph g starting at vertex s and continuing for a maximum of niter steps. Return a vector of vertices visited in order.

self_avoiding_walk(g, s, niter; rng=nothing, seed=nothing)

Perform a self-avoiding walk on graph g starting at vertex s and continuing for a maximum of niter steps. Return a vector of vertices visited in order.

eulerian(g::AbstractSimpleGraph{T}[, u::T]) --> T[]

Returns a Eulerian trail or cycle through an undirected graph g, starting at vertex u, returning a vector listing the vertices of g in the order that they are traversed. If no such trail or cycle exists, throws an error.

A Eulerian trail or cycle is a path that visits every edge of g exactly once; for a cycle, the path starts and ends at vertex u.

Optional arguments

• If u is omitted, a Eulerian trail or cycle is computed with u = first(vertices(g)).

all_simple_paths(g, u, v; cutoff)  --> Graphs.SimplePathIterator
all_simple_paths(g, u, vs; cutoff) --> Graphs.SimplePathIterator

Returns an iterator that generates all simple paths in the graph g from a source vertex u to a target vertex v or iterable of target vertices vs. A simple path has no repeated vertices.

The iterator's elements (i.e., the paths) can be materialized via collect or iterate. Paths are iterated in the order of a depth-first search.

If the requested path has identical source and target vertices, i.e., if u = v, a zero-length path [u] is included among the iterates.

Keyword arguments

The maximum path length (i.e., number of edges) is limited by the keyword argument cutoff (default, nv(g)-1). If a path's path length is greater than cutoff, it is omitted.

Examples

julia> g = complete_graph(4);

julia> spi = all_simple_paths(g, 1, 4)
SimplePathIterator{SimpleGraph{Int64}}(1 → 4)

julia> collect(spi)
5-element Vector{Vector{Int64}}:
 [1, 2, 3, 4]
 [1, 2, 4]
 [1, 3, 2, 4]
 [1, 3, 4]
 [1, 4]

We can restrict the search to path lengths less than or equal to a specified cut-off (here, 2 edges):

julia> collect(all_simple_paths(g, 1, 4; cutoff=2))
3-element Vector{Vector{Int64}}:
 [1, 2, 4]
 [1, 3, 4]
 [1, 4]