This paper is concerned with the partition of edges of the complete graph Kn and the complete multipartite graph K-m, ... ,-m into subgraphs isomor-phic to cycles. We show that Kn and K-m, ... ,m can be decomposed into certain families of cycles by defining a special decomposition which we call a root path decomposition.

This report deals with a parallel algorithmic technique that has proved to be very useful in the design of e cient parallel algorithms for several problems on undirected graphs. We describe this method for searching undirected graphs, called \open ear decomposition", and we relate this decomposition to graph biconnectivity. We present an e cient parallel algorithm for nding this decomposition a...

In a graph G, the distance from an edge e to a set F ⊆ E(G) is the vertex distance from e to F in the line graph L(G). For a decomposition of E(G) into k sets, the distance vector of e is the k-tuple of distances from e to these sets. The decomposition dimension dec(G) of G is the smallest k such that G has a decomposition into k sets so that the distance vectors of the edges are distinct. For ...

This report deals with a parallel algorithmic technique that has proved to be very useful in the design of eecient parallel algorithms for several problems on undirected graphs. We describe this method for searching undirected graphs, called \open ear decomposition", and we relate this decomposition to graph biconnectivity. We present an eecient parallel algorithm for nding this decomposition a...

Westudy some structural properties for tree-decompositions of 2-connected planar graphs that we use to improve upon the runtime of tree-decomposition based dynamic programming approaches for several NP-hard planar graph problems. E.g., we derive the fastest algorithm for Planar Dominating Set of runtime 3tw · nO(1), when we take the width tw of a given tree-decomposition as the measure for the ...

A graph G has tree-width at most w if it admits a tree-decomposition of width ≤ w. It is known that once we have a tree-decomposition of a graph G of bounded width, many NP -hard problems can be solved for G in linear time. For w ≤ 3 we give a lineartime algorithm for finding such a decomposition and for a general fixed w we obtain a probabilistic algorithm with execution time O(n log n+ n logn...

Modeling and understanding complex non-manifold shapes is a key issue in shape analysis and retrieval. The topological structure of a non-manifold shape can be analyzed through its decomposition into a collection of components with a simpler topology. Here, we consider a decomposition of a non-manifold shape into components which are almost manifolds, and we present a novel graph representation...

In this thesis we study the geometry and topology of Riemannian 3-orbifolds which are locally volume collapsed with respect to a curvature scale. Our main result is that a sufficiently collapsed closed 3-orbifold without bad 2-suborbifolds satisfies Thurston’s Geometrization Conjecture. We also prove a version of this result with boundary. Kleiner and Lott indepedently and simultanously proved ...

Given two r-graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. The minimum number of parts in an H-decomposition of G is denoted by φH(G). By a 2-edge-decomposition of an r-graph we mean an H-decomposition for any fixed r-graph H with exactly 2 edges. In the special case where the two edges of...

We introduce nondeterministic graph searching with a controlled amount of nondeterminism and show how this new tool can be used in algorithm design and combinatorial analysis applying to both pathwidth and treewidth. We prove equivalence between this game-theoretic approach and graph decompositions called q-branched tree decompositions, which can be interpreted as a parameterized version of tre...