Vertices of degree k in edge-minimal, k-edge-connected graphs

# Vertices of degree k in edge-minimal, k-edge-connected graphs

Carl Kingsford C. Kingsford, Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA  and  Guillaume Marçais G. Marçais, Program in Applied Mathematics & Statistics and Scientific Computation, University of Maryland, College Park, MD, USA
July 14, 2019
C.K. was partially supported by NSF grant IIS-0812111.

Halin  showed that every edge-minimal, -vertex-connected graph has a vertex of degree . In this note, we prove the analogue to Halin’s theorem for edge-minimal, -edge-connected graphs:

###### Theorem 1.

Let be an edge-minimal, -edge-connected graph. Then there are two nodes of degree in .

To prove Theorem 1, we first establish a link between edge-minimal, -edge-connected graphs and exactly -edge-connected graph  (Definition 7 and Proposition 9). The theorem is proved in the case of being an exactly -edge-connected graph (Proposition 16) and then transfered to edge-minimal graphs. Throughout, all graphs considered are multigraphs and all sets are multisets.

## The edge-connectivity equivalence relation.

###### Proposition 2.

The edge version of Menger’s theorem holds for multigraphs.

###### Proof.

Any multi-edge in the graph can be replaced by a length path to get a graph . There is then an obvious bijection between the paths in the original multigraph and the resulting graph . The result of the Menger theorem on can be immediatly applied to .∎

###### Definition 3.

Let be the relation between nodes and that holds if or there are edge-disjoint paths between and .

###### Proposition 4.

is an equivalence relation.

###### Proof.

is by definition reflexive and symmetric. Let be a graph, let satisfy and . If , or , then transitivity is obvious. Suppose that , and are distinct vertices and let be an edge set of cardinality . There are edge-disjoint paths from to in so, and are still connected in and so are and . So we have a path in . By Menger’s theorem, the set of paths between and in has cardinality at least and , and is transitive.∎

###### Proposition 5.

There are at most edge-disjoint paths between two equivalency classes of .

###### Proof.

Suppose we have edge-disjoint paths, , , between and , two distinct equivalency classes of in graph . Let , and , be the endpoints in and respectevily of edge-disjoint paths. (Note that the are not all necessary distinct. This is true for the as well.) Let be a set of edges. Then, in , at least one path, say , was not disconnected. Because , and are not disconnected. Similarly, and are not disconnected. So we have a path . By Menger’s theorem, , which is a contradiction. ∎

## Relationship between exact connectivity and minimality.

###### Proposition 6.

Let be a -edge-connected graph. is edge minimal if and only if for any adjacent vertices , there are at most edge disjoint paths.

###### Proof.

Let such that there are edge-disjoint paths. In , there are at least edge-disjoint paths. Let and be any two vertices, not necessarily distinct from and . There are edge-disjoint paths in . So, depending on whether or not edge is on one of these paths, in there are either edge-disjoint paths, or there are edge-disjoint paths and a path edge disjoint from the paths. There is a similar situation in between , and . Let be a set of distinct edges of . In , there is a or path, a or path and a path. Hence there is a path and is not a separating set. By Menger’s theorem, is -edge-connected, and is not edge minimal. Conversely, suppose that for any edge there are at most edge-disjoint paths. Then in there are at most edge disjoint paths. So is edge minimal.∎

###### Definition 7.

A graph is called exactly -edge-connected if there are exactly edge disjoint paths between any two nodes .

###### Definition 8.

Let be a -edge-connected graph. Define to be a graph where the vertices are the equivalency classes of on and there is an edge for every edge with and .

###### Proposition 9.

If is an edge-minimal, -edge-connected graph then is not trivial and is exactly -edge-connected.

###### Proof.

If is edge minimal, it is not -edge-connected, so has more than one equivalency class, and is not trivial. is -edge-connected like . By Proposition 5, it is exactly -edge-connected.∎

###### Proposition 10.

Let be an edge-minimal, -edge-connected graph, and let be an equivalence class of . Then, for any , .

###### Proof.

Let . By Proposition 6, if , then is not an edge in . So every neighbor of is not in and by construction . ∎

## Proof of Theorem 1.

###### Definition 11.

An edge cut of a graph is called trivial if one of the components of is the trivial graph.

###### Definition 12.

A -regular graph is a graph where all vertices have the same degree . A quasi -regular graph is a graph where at most one vertex has a degree different than .

###### Lemma 13.

An exactly -edge connected graph which has only trivial cuts is quasi -regular.

###### Proof.

Suppose there exists two vertices and of degree greater than . There exists a minmum cut separating and and this cut cannot be trivial.∎

###### Definition 14 (Vertex splitting).

Let be an exactly -edge connected graph and be a non-trivial minimum cut. Construct and by adding two new vertices and attached respectively to and by new edges to the vertices adjacent to . Formally, let and

 E1 = (V1×V1∩E)∪{(x1,ui):1≤i≤k} E2 = (V2×V2∩E)∪{(x2,vi):1≤i≤k}.

The pair is called a vertex splitting of with respect to .

###### Proposition 15.

Let be an exactly -edge-connected graph, and let be a non-trivial minimum cut. and obtained from vertex splitting with respect to are exactly -edge-connected.

###### Proposition 16.

Let be an exactly -edge-connected graph. Then there are two vertices of degree in .

###### Proof.

We proceed by induction on the number of non-trivial minimum cuts in . If has no non-trivial minimum cuts, then it is quasi -regular and has at least 2 nodes of degree . Let be a non-trivial minimum cut and let and be the vertex splitting graphs induced by . Call and the new vertices (). Suppose were a non-trivial minimum cut in . Construct a corresponding non-trivial minimum cut in by changing any edge used by that is adjacent to to the corresponding edge in . So to any non-trivial minimum cut of or corresponds a distinct non-trivial minimum cut in . But no non-trivial cut in or corresponds to the cut (it would be a trivial cut in and ). So both and have fewer non-trivial minimum cuts than . By induction and have 2 nodes of degree , including and . So, has the same vertices as and , except for and , with the same degree. Hence it has 2 nodes of degree . ∎

###### Proof of Theorem 1.

By Proposition 16, has two vertices of degree . Let be an equivalence class of . If contains two distinct vertices and of , then there are at least edge disjoint paths in and both and have a degree . By Proposition 10, also has degree in . Hence, the vertices of degree in correspond to equivalence classes that must each contain at most one vertex of of degree . Therefore, has two vertices of degree . ∎

## References

•  R. Halin. A theorem on -connected graphs. Journal of Combinatorial Theory, 7:150–154, 1969.
•  Carl Kingsford and Guillaume Marçais. A synthesis for exactly 3-edge-connected graphs. Submitted to FOCS 2009, May 2009, arXiv:0905.1053 [math.CO].

You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters   