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Making multigraphs simple by a sequence of double edge swaps

https://doi.org/10.1016/j.disc.2021.112328Get rights and content

Abstract

We show that any loopy multigraph with a graphical degree sequence can be transformed into a simple graph by a finite sequence of double edge swaps with each swap involving at least one loop or multiple edge. Our result answers a question of Janson motivated by random graph theory, and it adds to the rich literature on reachability of double edge swaps with applications in Markov chain Monte Carlo sampling from the uniform distribution of graphs with prescribed degrees.

Introduction

We will consider different classes of undirected graphs, the most general being loopy multigraphs where both multiple edges and multiple loops are allowed. Specifically, we are interested in graphs where each vertex has a prescribed degree, the degree of a vertex being the number of stubs (half-edges) attached to it (so the contribution from a loop is two). The list of the degrees of all vertices, sorted in weakly decreasing order, is called the degree sequence of the graph, and a weakly decreasing sequence is said to be graphical if it is the degree sequence of some simple graph (no loops and no multiple edges). The most popular basic graph operation that preserves the degree sequence is the replacement of any two edges (v1,v2) and (v3,v4) by (v2,v3) and (v4,v1). This is called a double edge swap and was first introduced by Petersen [12]. It has been reinvented several times and has many alternative names in the literature [11]: degree-preserving rewiring, chequerboard swap, tetrad or alternating rectangle.

The main motivation for our work comes from the theory of random graphs. There is a simple direct method of generating a uniformly random stub-labelled (where the stubs have identity) loopy multigraph with prescribed degrees: Attach the prescribed number of stubs to each vertex, then choose a random matching of all stubs. This is called the configuration model and was introduced by Bollobás 1980 [1]. The simplicity of the method makes it very useful for theoretical analyses of random graphs, but in many applications one wants to study simple graphs rather than multigraphs. There are several possible solutions to this issue. Sometimes it is possible to simply condition the random loopy multigraph from the configuration model on the event that it is a simple graph. This yields a uniform distribution of simple graphs with the given degree sequence. Recently, Janson [8] proposed another method, the switched configuration model, where the random loopy multigraph is transformed into a simple graph by a sequence of random double edge swaps. Each swap is required to have the property that at least one of the two swapped edges is a loop or a multiple edge. The resulting distribution on simple graphs is not exactly uniform, but for a certain class of degree sequences Janson showed that it is asymptotically uniform in the sense that the total variation distance to the uniform distribution tends to zero when the number of vertices goes to infinity. Motivated by his construction, he posed the following question to us in person:

Question 1

Can any loopy multigraph with a graphical degree sequence be transformed into a simple graph by a finite sequence of double edge swaps involving at least one loop or multiple edge?

In this paper, we answer the question affirmatively. In fact, we show a stronger statement that Jansson conjectured in [8, Remark 3.4], namely that it is always possible to reach a simple graph even if an evil person chooses which loop or multiple edge should be involved in each double edge swap.

Our result adheres to a rich literature of reachability of double edge swaps, a topic that has an important application in the context of Markov chain Monte Carlo sampling; see Fosdick et al. [11] for a comprehensive discussion. In the simplest case, we want to sample from the uniform distribution of all graphs (of some class) with prescribed degrees. Basically, one starts with any graph with the given degrees and performs random double edge swaps for a while; the stationary distribution is uniform. (Exactly how the random double edge swaps should be chosen depends on the class of graphs and the type of labelling of the graph, see [11].) To show uniformity, one has to verify that the Markov chain satisfies three conditions:

  • (i)

    that the transition matrix of the chain is doubly stochastic,

  • (ii)

    that the chain is irreducible,

  • (iii)

    and that the chain is aperiodic.

The irreducibility condition means that for any pair of graphs G and G with the same degree sequence there is a sequence of double edge swaps that transforms G to G. If this is true or not depends on the particular class of graphs we are interested in. It is true for simple graphs [2], [4], [6], connected simple graphs [13], 2-connected simple graphs [14], loop-free multigraphs [7], simple-loopy multigraphs (multiple edges and simple loops) [9] and loopy multigraphs [3], but not for simple-loopy simple graphs (simple edges and simple loops) [10] and loopy simple graphs (where multiple loops are allowed but no other multiple edges) [9].

Note how our result differs from that of Eggleton and Holton [3]. While they show that any loopy multigraph can be transformed into any other loopy multigraph with the same degree sequence by a sequence of double edge swaps, we show that this can be accomplished with admissible swaps only, where a swap is admissible if it involves at least one loop or multiple edge. In the situation where Janson posed Question 1, this condition is natural since the goal is to reach a simple graph. In applications, when using the switched configuration model to sample from an approximately uniform distribution of simple graphs with a given degree sequence, one wants to obtain a simple graph by as few double edge swaps as possible, so swapping away “bad” edges is essential for the efficiency of this method.

The paper is organized as follows. First, in Section 2 we fix the notation and recall the Erdős–Gallai theorem. In Section 3 we present our results and in Sections 4 Proof of, 5 Alternative proof of we prove them. Finally, in Section 6 we discuss some open questions.

Section snippets

Notation and prerequisites

The terminology on multigraphs is not standardized, so let us start by defining it. Fig. 1 shows some examples.

A loop is an edge connecting a vertex to itself. A loopy multigraph is an undirected graph where loops are allowed and where there might be multiple edges between the same pair of vertices and multiple loops at the same vertex.

A loop-free multigraph is a loopy multigraph without loops.

An edge is said to be simple if it has multiplicity one and is not a loop, and a graph is simple if

Results

Our main result is the following.

Theorem 2

Any loop-free multigraph whose degree sequence is graphical can be transformed into a simple graph by a finite sequence of admissible double edge swaps.

Fig. 3 shows an example.

Let us state a simple consequence of Theorem 2:

Theorem 3

Any loopy multigraph whose degree sequence is graphical can be transformed into a simple graph by a finite sequence of admissible double edge swaps.

Proof

Consider a loopy multigraph whose degree sequence is graphical. If there is a loop at some

Proof of Theorem 4

First, we need yet another game to play. Let H be a loop-free multigraph on a vertex set V. In the loop-free multigraph game with target H, starting from a loop-free multigraph G on V such that every vertex has the same degree in G as in H, in each move the Devil chooses any edge e in G such that G has more edges than H of the same type as e (that is, with the same endpoints), and then the Angel chooses any edge e in G not incident to e and performs a double edge swap on e and e in G. The

Alternative proof of Theorem 2

Note that Theorem 2 follows directly from Theorem 4. Our proof of Theorem 4 uses the fact that the degree sequence is graphical to choose a simple graph H as a target. In this section we present an alternative proof of Theorem 2 which does not rely on such a choice. In fact, the only way it exploits the graphicality of the degree sequence is via the inequalities guaranteed by the easy “only if” direction of the Erdős–Gallai theorem. With this in mind, the harder “if” direction of the

Open questions

In this paper, we have focused on the existence of a sequence of admissible double edge swaps that makes a graph simple. For further research, it seems natural to ask about the length of such a sequence.

Question 2

What is the minimum number of admissible double edge swaps needed to transform a given loopy multigraph G with a graphical degree sequence into a simple graph?

As we saw in Fig. 3, it is not always possible to decrease the number of multiple edges by a double edge swap. On the other hand, some

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

I am very grateful to the anonymous referees. One of them suggested an approach that led to the proof of Theorem 4 (which was a conjecture in an earlier version of the paper).

This work was mostly conducted while the author was a researcher as Uppsala University. It was supported by the Knut and Alice Wallenberg Foundation .

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