CHAIN DECOMPOSITIONS OF 4-CONNECTED GRAPHS

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)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(vertical bar V (G)vertical bar(2)vertical bar E (G)vertical bar) algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be improved to O(vertical bar V (G)vertical bar(3)). This decomposition will be used to find four independent spanning trees in a 4-connected graph.194848880Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)NSF [DMS9970527, DMS0245530]NSA [MDA904-03-1-0052]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPq [200611/00-3]NSF [DMS9970527, DMS0245530]NSA [MDA904-03-1-0052

Finding four independent trees

Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2-connected graph. Cheriyan and Maheshwari gave an O(vertical bar V vertical bar(2)) algorithm for finding three independent spanning trees in a 3-connected graph. In this paper we present an O(vertical bar V vertical bar(3)) algorithm for finding four independent spanning trees in a 4-connected graph. We make use of chain decompositions of 4-connected graphs.3551023105

Subdivisional spaces and graph braid groups

We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we decompose $X$ and show that the homology of the configuration spaces of $X$ is computed by the derived tensor product of the Morse complexes of the pieces of the decomposition, an analogue of the monoidal excision property of factorization homology. Applying this theory to the configuration spaces of a graph, we recover a cellular chain model due to \'{S}wi\k{a}tkowski. Our method of deriving this model enhances it with various convenient functorialities, exact sequences, and module structures, which we exploit in numerous computations, old and new.Comment: 71 pages, 15 figures. Typo fixed. May differ slightly from version published in Documenta Mathematic

On the phase transitions of graph coloring and independent sets

We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph's chromatic number and independence number. We provide empirical evidence that, along a sequence of increasingly denser random graphs, the fraction of acyclic orientations that are `shortest' peaks when the chromatic number increases, and that such maxima tend to coincide with locally easiest instances of the problem. Similar evidence is provided concerning the `widest' acyclic orientations and the independence number