Connected resolving decompositions in graphs

Connected Resolving Partitions in Unicyclic Graphs

A k-partition Π = {S1,S2, . . . ,Sk} of V (G) is resolving if for every two distinct vertices u and v of a connected graph G, there is a set Si in Π so that the minimum distance between u and a vertex of Si is different from the minimum distance between v and a vertex of Si. A resolving partition Π is said to be connected if each subgraph < Si > induced by Si (1 ≤ i ≤ k) is connected in G. In t...

Chain Decompositions of 4-Connected Graphs

In this paper we give a decomposition of a 4-connected graph G into nonseparating chains, which is similar to an ear decomposition of a 2-connected graph. We also give an O(|V (G)||E(G)|) algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be improved to O(|V (G)|). This decomposition will be used to find four independent spanning trees in a 4co...

On 4-connected graphs without even cycle decompositions

Decompositions of highly connected graphs into paths of length five

We prove that a 171-edge-connected graph has an edgedecomposition into paths of length 3 if and only its size is divisible by 3. It is a long-standing problem whether 2-edge-connectedness is sufficient for planar triangle-free graphs, andwhether 3-edge-connectedness suffices for graphs in general. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 286–292, 2008 Keyword: path-decompositions

Resolving Decompositions for Polynomial Modules

We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of free resolutions. It provides a unifying framework for recent results of the authors for different types of bases.