ÐIn this work, we show that the standard graph-partitioning-based decomposition of sparse matrices does not reflect the actual communication volume requirement for parallel matrix-vector multiplication. We propose two computational hypergraph models which avoid this crucial deficiency of the graph model. The proposed models reduce the decomposition problem to the well-known hypergraph partition...

In this paper, we design the first linear-time algorithm for computing the prime decomposition of a digraph G with regard to the cartesian product. A remarkable feature of our solution is that it computes the decomposition of G from the decomposition of its underlying undirected graph, for which there exists a linear-time algorithm. First, this allows our algorithm to remain conceptually very s...

We study the problem of decomposing the vertex set V of a graph into two nonempty parts V1, V2 which induce subgraphs where each vertex v ∈ V1 has degree at least a(v) inside V1 and each v ∈ V2 has degree at least b(v) inside V2. We give a polynomialtime algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition, and gives such a decomposition if it exists. This...

Let L = {H1, H2, . . . ,Hr} be a family of subgraphs of a graph G. An L-decomposition of G is an edge-disjoint decomposition of G into positive integer αi copies of Hi, where i ∈ {1, 2, . . . , r}. Let Ck, Pk and Sk denote a cycle, a path and a star with k edges, respectively. In this paper, we prove that a balanced complete bipartite graph with 2n vertices has a {Ck, Pk, Sk}-decomposition if a...

Network analysis is an important step in the understanding of complex systems studied in various areas such as biology, geography or sociology. This thesis focuses on the problems related to the decomposition of those networks when they are modeled by graphs. Graph decomposition methods are useful for data compression, community detection or network visualisation. One possible decomposition is ...

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with ( n+1 2 ) edges admits an edge decomposition G = H1 ⊕ · · · ⊕Hn such that Hi has i edges and is isomorphic to a subgraph of Hi+1, i = 1, . . . , n−1. We show that every bipartite graph G with ( n+1 2 ) edges such that the degree sequence d1, . . . , dk of one of the stable sets satisfies di ≥ n − i + 2, 1 ≤ i ...

A graph is Berge if no induced subgraph of it is an odd cycle of length at least five or the complement of one. In joint work with Robertson, Seymour, and Thomas we recently proved the Strong Perfect Graph Theorem, which was a conjecture about the chromatic number of Berge graphs. The proof consisted of showing that every Berge graph either belongs to one of a few basic classes, or admits one o...

Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these graphs is from Nakano et al. (J. Comp. Sci. Tech., 2007), who have proven that the number of distancehereditary graphs on n vertices is bounded by 2d3.59ne. In t...

Dilworth s theorem states a duality relation between minimum chain decompositions of a directed acyclic graph and maximum antichains We generalize the theorem to apply when the chains of the decomposition are required to contain the chains of an initial decomposition We show that duality obtains precisely when an associated undirected graph is perfect We apply this result to a vehicle routing a...

We show that a cooperative game may be decomposed into a sum of component games, one for each player, using the combinatorial Hodge decomposition on a graph. This decomposition is shown to satisfy certain efficiency, null-player, symmetry, and linearity properties. Consequently, we obtain a new characterization of the classical Shapley value as the value of the grand coalition in each player’s ...