Minimum Degree and Graph Minors

Gasper Fijavz 1,2

Faculty of Computer and Information ScienceUniversity of Ljubljana

Ljubljana, Slovenia

David R. Wood 3,4

Departament de Matematica Aplicada IIUniversitat Politecnica de Catalunya

Barcelona, Spain

Abstract

A graph G is a minor minimal minimum degree graph (MMMD) if δ(H) < δ(G)for every proper minor H of G. We (i) determine all complete multipartite MMMDgraphs and show that (ii) every small k-regular graph is a MMMD graph. Intuitivelyit seems that MMMD graphs are highly connected. Countering that (iii) we showthat MMMD graphs may have a rich block-structure.

Keywords: graph minors, forbidden minors, minimum degree

1 Supported in part by the Ministry of Higher Education, Science and Technology of Slove-nia, Research Program P1-0297.2 Email:gasper.fijavz@fri.uni-lj.si3 Supported by a Marie Curie Fellowship from the European Commission under con-tract MEIF-CT-2006-023865, and by the projects MEC MTM2006-01267 and DURSI2005SGR00692.4 Email:david.wood@upc.edu

Electronic Notes in Discrete Mathematics 31 (2008) 79–83

1571-0653/$ – see front matter © 2008 Elsevier B.V. All rights reserved.

www.elsevier.com/locate/endm

doi:10.1016/j.endm.2008.06.013

1 Introduction

The theory of graph minors developed by Robertson and Seymour [10] is one ofthe most important in graph theory influencing many branches of mathematicsand beyond. The concept of a graph minor itself dates back to Wagner [11]connecting topological and structural properties of graphs on one hand withthe combinatorial properties.

Minor-monotone graph parameters, graph genus and many other examplessuch as Colin de Verdiere’s parameter μ [3], serve as first indicators on thetopological and structural complexity of a graph. On the other hand, a graphparameter, even if describing topological structure of a graph, may not workwell with graph minors. Observe the crossing number of a graph, for example.

Choose an arbitrary integer k. If ϕ is a minor-monotone graph param-eter, then the class of graphs G satisfying ϕ(G) < k is closed under takingminors, and its set of forbidden minors is finite by the graph minor theoremof Robertson and Seymour [10].

There are, however, a couple of ways to transform a graph parameter ϕinto a related minor-monotone one. For a fixed graph G, we can compute themaximal value of ϕ(H), where we let H run over all minors of G. Alterna-tively, we can compute the minimal value of ϕ(H ′), where H ′ is an arbitrarygraph, which contains G as a minor [4]. The minor-crossing number [1,2] isan example of the latter approach.

Let Dk denote the class of graphs G such that every minor of G has min-imum degree δ at most k. Dk is closed under taking minors and we let Dk

denote the set of forbidden minors for Dk. It is easy to observe that G ∈ Dk

if and only if max{δ(H) ; H is a minor of G} ≤ k. Also G ∈ Dk if and only ifδ(G) ≥ k + 1 and every proper minor of G has min-degree at most k.

Further let Ck denote the class of graphs G such that every minor of Ghas (vertex) connectivity κ at most k. Ck is a minor closed class and Ck shalldenote its set of forbidden minors.

This abstract focuses on graphs in Dk and the relations between Dk andCk. Since κ(G) ≤ δ(G) we have Ck ⊆ Dk for every integer k.

2 Results and questions

Let us first state some small examples, D1 = C1 = {K3}, D2 = C2 = {K4},and D3 = C3 = {K5, K2,2,2}. The sets Dk and Ck, for k ≥ 4, have not been

determined yet. It has been conjectured [4] that |C4| = 6 and computing therest seems to be a tremendeously hard task. Hence it may be worthwile to

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look for graphs in Ck or Dk that have additional structural properties [6,7,5].

It is easy to show that every 4-connected planar graph contains K2,2,2 asa minor, which, together with Wagner’s decomposition theorem [11] implies

that C3 = {K5, K2,2,2}. The fact that every graph G with δ(G) ≥ 4 contains aK5 or K2,2,2 minor is more challenging. It has often been attributed to Halinand Jung [8] but it does not appear in the text of the paper.

Mader [9] exhibited a 2-connected graph G12 on 12 vertices with δ(G12) = 5

which (i) belongs to D4 and (ii) does not contain a 5-connected minor, showingthat Dk �= Ck for k ≥ 4. The smallest example has one vertex less:

Theorem 2.1 Every graph G with δ(G) ≥ 5 and at most 10 vertices containsa 5-connected minor, and there exists a graph G11 on 11 vertices with δ(G11) =5 which does not contain a 5-connected minor.

Clearly Kk+2 ∈ Dk and there exist several tools describing how to obtainnew graphs in Dk+1 from graphs in Dk. It is also not difficult to see that everyk +1-connected graph in Dk belongs to Ck. But the converse may not be true.Is Ck equal to the set of k + 1-connected graphs from Dk? Alternatively, isCk ⊆ Dk?

We show that Dk contains all small k + 1 regular graphs.

Theorem 2.2 Every k + 1-regular graph G with less than 43(k + 2) vertices is

in Dk.

The bound in Theorem 2.2 is tight. For every k ≡ 1 (mod 3) there exists

a k + 1-regular graph G on 43(k + 2) vertices that is not in Dk.

It seems plausible that graphs in Dk are well connected. But alreadyMader’s example G12 shows that its connectivity need not be larger than 2.By deleting a pair of matchings (of suitable sizes) in a pair of Kk+2s and

joining one new vertex to the matched vertices one can obtain a graph in Dk

that has connectivity equal to 1. Even more, graphs in Dk can almost havean arbitrary block structure:

Theorem 2.3 Let T an arbitrary tree with vertex bipartition V1 ∪ V2 so thatvertices in V1 have degree ≤ 2 and vertices in V2 have degree ≥ 2. There existsan integer d and a graph G such that

(i) G is d-regular,

(ii) T is the block decomposition tree of G,

(iii) d ≤ 4|E(T )|, and

(iiii) G ∈ Dd−1.

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The degree condition on V1 forces that blocks in G have at most twocutvertices. A single cutvertex may be in many blocks, however.

As a direct corollary of Theorem 2.3 we can show that the maximal diam-eter of graphs in Dk is of order at least k. By a bit more careful constructionwe have for even k found k + 1-regular graphs in Dk with diameter 3k − 1.What is the maximal possible diameter of graphs in Dk?

Proving an upper bound on the diameter may be the crucial step towardsproving that Dk is finite without using the graph minor theorem.

The class of complete multipartite graphs is closed under edge contractions.Note that for a complete multipartite graph G we have δ(G) = κ(G). These

properties enable us to characterize complete multipartite graphs in Dk an Ck.

Theorem 2.4 For all k ≥ 1, a complete multipartite graph G is in Dk (or

Ck) if and only if for some b ≥ a ≥ 1 and p ≥ 2

G = Ka,b, . . . , b︸ ︷︷ ︸

p

such that k + 1 = a + (p − 1)b and if p = 2 then a = b.

Graphs embeddable in a fixed surface form a contraction closed class ofgraphs. Yet there are very few graphs of min-degree (or connectivity) ≥ 7 ifthe surface is fixed. Is there another sensible class of graphs G, which is closedunder edge contractions, so that Dk ∩ G or Ck ∩ G can be determined?

References

[1] Bokal, D., Fijavz, G., and Mohar, B., The minor crossing number, SIAM J.Discrete Math. 20 (2006), 344–356.

[2] Bokal, D., Fijavz, G., and Wood, D. R., The Minor Crossing Number of Graphswith an Excluded Minor, Electron. J. Combin. 15:R4 (2008).

[3] Colin de Verdiere, Y., Sur un nouvel invariant des graphes et un critere deplanarite, J. Combin. Theory Ser. B 50 (1990), 11–21.

[4] Fijavz, G., “Graph minors and connectivity,” Ph.D. thesis, University ofLjubljana, 2001 (in Slovene).

[5] Fijavz, G.,, Minor-minimal 6-regular graphs in the Klein bottle, European J.Combin. 25 (2004), 893–898.

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[6] Fijavz, G., Contractions of 6-connected toroidal graphs, J. Combin. Theory Ser.B 97 (2007), 553–570.

[7] Fijavz, G., Minor-minimal 5-connected projective-planar graphs, submitted.

[8] Halin, R., and Jung, H. A., Uber Minimalstrukturen von Graphen, insbesonderevon n-fach zusammenhangenden Graphen, Math. Ann. 152 (1963), 75–94.

[9] Mader, W., Homomorphiesatze fur Graphen, Math. Ann. 178 (1968), 154–168.

[10] Robertson, N., and Seymour, P. D., Graph minors I-XX, J. Combin. TheorySer. B, (1983 – 2004).

[11] Wagner, K., Uber eine Eigenschaft der ebene Komplexe, Math. Ann. 114 (1937),570–590.

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