# Enumeration of multigraph DFS

Posted on March 1, 2017

This is written in Literate Haskell.1

``````module Enumerating.DFS where

import Test.Feat.Enumerate
import Test.Feat.Access``````

## A representation problem

What is a good way to implement graphs? The problem with standard methods, i.e., adjacency lists and matrices, is that they are badly structured.

In contrast, trees can be defined inductively: a (rooted) tree is a vertex linked via edges to some (or no) smaller trees called its children. This is a good representation because we can easily define operations on trees recursively.

``````data Tree = Vertex [Edge]
type Edge = Tree``````

To represent more general graphs, we can start with a tree like that, and encode the remaining edges specially. One simple type of edge is the back edge: linking a vertex to one of its ancestors. Since there is only one way up, a back edge can simply be encoded as an integer counting the number of generations separating the vertex from its ancestor. This is just like de Bruijn indices in lambda terms, and here every vertex is a binder.

``````data TreeB = VertexB [EdgeB]
data EdgeB = TreeB TreeB | BackB Integer``````

## DFS Trees

From any undirected graph, we may restructure it as a tree with back edges by performing a DFS. In fact, the above type naturally represents DFS of undirected multigraphs, where a vertex can have many back edges pointing to the same ancestor. Different DFS can be obtained from the same multigraph, depending on the chosen root, and the order in which edges are crossed.

Thanks to that inductive definition, we can enumerate such trees. We must however be careful to avoid back edges pointing past the root of the tree. This notion of well-formedness is formalized by generalizing it with a context, which is the number of ancestors of the current root.

The `testing-feat` package2 defines a type `Enumerate` to efficiently enumerate combinatorial species, i.e., sets of things classified by some notion of size, such that there is finitely many things of a given size.

``type Species = Enumerate``

We define inductively the combinatorial species of rooted multigraphs with a context of `n` ancestors, with size measured as the number of edges.

``````treeB :: Int -> Species TreeB
treeB n = VertexB <\$> treeBChildren n``````

The root costs nothing, and we must then enumerate the species of lists of children. We distinguish two subspecies of which this is the sum: the species of no children and the species of some children.

``````treeBChildren :: Int -> Species [EdgeB]
treeBChildren n = noChildren <> someChildren n``````

The species of no children contains a single object of size 0: the empty list. In `treeB` this corresponds to the graph with no vertices.

``````noChildren :: Species [EdgeB]
noChildren = pure []``````

### Making the children pay

The species of some children corresponds to non-empty lists. They contain at least one element `child`, followed by an arbitrary list `children`. The first element will use one additional edge to connect to the root; the `pay` combinator adds one to the size of each element in a species.

``````someChildren :: Int -> Species [EdgeB]
someChildren n = pay
(liftA2
(\ child children -> child : children)
(treeBChild n)
(treeBChildren n))``````

Now, we may obtain a child by following an edge which belong to one of two types. A tree edge `(TreeB <\$> treeB (n + 1))` links the current root as the parent of a new root, which thus has one more ancestor in addition to the previous ones. A back edge links to one of the `n` ancestors. The auxiliary `backEdges n` defines the species with `n` elements `[BackB 0 .. BackB (n - 1)]` all of size `0`; the size of the back edge was already `pay`-d by `someChildren`.

``````treeBChild :: Int -> Species EdgeB
treeBChild n = (TreeB <\$> treeB (n + 1)) <> backEdges n
where
backEdges n = fromParts [Finite (toInteger n) (BackB . fromInteger)]``````

This concludes the definition of `treeB`. Let us enumerate these graphs (with no implicit ancestors, i.e., with context `n = 0`).

``````count :: Species a -> [Integer]
count s = [cardinal | (cardinal, _) <- valuesWith s]``````

In a few seconds we get:

``````oeisA258173 = count (treeB 0)

-- [1,1,3,12,58,321,1975,13265,96073,743753,6113769,53086314,
-- 484861924,4641853003,46441475253,484327870652,5252981412262...``````

At the time of writing, the conjecture that this class of graphs is enumerated by this sequence is not mentioned in the OEIS. I suspect this also is not too hard to prove.

March 2, 2017. Update:

Thanks to Antti Karttunen for the proof3 that these sequences are indeed the same.