A Note on Contraction Degeneracy

The parameter contraction degeneracy — the maximum minimum degree over all minors of a graph — is a treewidth lower bound and was first defined in [3]. In experiments it was shown that this lower bound improves upon other treewidth lower bounds [3]. In this note, we examine some relationships between the contraction degeneracy and connected components of a graph, blocks of a graph and the genus of a graph. We also look at chordal graphs, and we study an upper bound on the contraction degeneracy. A data structure that can be used for algorithms computing the degeneracy and similar parameters, is also described.