Edge Decomposition Research Articles

Data Mining Techniques in Medical Informatics

Data Mining Techniques in Medical Informatics

Edge decompositions into two kinds of graphs

Let H be an arbitrary graph and let K 1 , 2 be the 2-edge star. By a { K 1 , 2 , H } -decomposition of a graph G we mean a partition of the edge set of G into subsets inducing subgraphs isomorphic to K 1 , 2 or H . Let J be an arbitrary connected graph of odd size. We show that the problem to decide if an instance graph G has a { K 1 , 2 , H } -decomposition is NP-complete if H has a component of an odd size and H ≠ p K 1 , 2 ∪ q J , where p K 1 , 2 ∪ q J is the disjoint union of p copies of K 1 , 2 and q copies of J . Moreover, we prove polynomiality of this problem for H = q J .

Timestamping messages and events in a distributed system using synchronous communication

Determining order relationship between events of a distributed computation is a fundamental problem in distributed systems which has applications in many areas including debugging, visualization, checkpointing and recovery. Fidge/Mattern’s vector-clock mechanism captures the order relationship using a vector of size N in a system consisting of N processes. As a result, it incurs message and space overhead of N integers. Many distributed applications use synchronous messages for communication. It is therefore natural to ask whether it is possible to reduce the timestamping overhead for such applications. In this paper, we present a new approach for timestamping messages and events of a synchronously ordered computation, that is, when processes communicate using synchronous messages. Our approach depends on decomposing edges in the communication topology into mutually disjoint edge groups such that each edge group either forms a star or a triangle. We show that, to accurately capture the order relationship between synchronous messages, it is sufficient to use one component per edge group in the vector instead of one component per process. Timestamps for events are only slightly bigger than timestamps for messages. Many common communication topologies such as ring, grid and hypercube can be decomposed into $${\lceil N/2 \rceil}$$ edge groups, resulting in almost 50% improvement in both space and communication overheads. We prove that the problem of computing an optimal edge decomposition of a communication topology is NP-complete in general. We also present a heuristic algorithm for computing an edge decomposition whose size is within a factor of two of the optimal. We prove that, in the worst case, it is not possible to timestamp messages of a synchronously ordered computation using a vector containing fewer than $${2 \lfloor N/6 \rfloor}$$ components when N ≥ 2. Finally, we show that messages in a synchronously ordered computation can always be timestamped in an offline manner using a vector of size at most $${\lfloor N/2 \rfloor}$$ .

Tree edge decomposition with an application to minimum ultrametric tree approximation

A k-decomposition of a tree is a process in which the tree is recursively partitioned into k edge-disjoint subtrees until each subtree contains only one edge. We investigated the problem how many levels it is sufficient to decompose the edges of a tree. In this paper, we show that any n-edge tree can be 2-decomposed (and 3-decomposed) within at most ⌈1.44 log n⌉ (and ⌈log n⌉ respectively) levels. Extreme trees are given to show that the bounds are asymptotically tight. Based on the result, we designed an improved approximation algorithm for the minimum ultrametric tree.

A new two-variable generalization of the chromatic polynomial

We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.

Cycle decompositions of crowns

For an integer n⩾3, the crown S n 0 is defined to be the graph with vertex set { x 0, x 1,…, x n−1 , y 0, y 1,…, y n−1 } and edge set {x iy j : 0⩽i,j⩽n−1, i≠j} . In this paper we give some sufficient conditions for the edge decomposition of the crown into isomorphic cycles.

On Edge Decompositions of Posets

An edge decomposition of a poset P is a collection of chains such that every pair of elements of which one covers the other belongs to exactly one chain. We consider this and the related notion of the line poset L(P) which consists of pairs of adjacent elements of P so that (x⋖y)<L(P) (x'⋖y') iff y ≤P x'. We present some min-max type results on path-cycle partitions of digraphs which are applicable to poset decompositions. Providing an explicit construction we show that the lattice of the subsets of an n-set admits an edge decomposition into symmetric chains. We demonstrate a few applications of this decomposition. Also, a characterisation of line posets is given.

The reconstruction of separable graphs with 2-connected trunks that are series-parallel networks

A 2-connected graph has a cleavage unit-virtual edge decomposition which is due to Tutte [8]. Cleavage units are either polygons, bonds (planar duals to polygons) or 3-connected simple graphs. When all cleavage units are polygons or bonds, such graphs are called series-parallel networks. The Ulam graph reconstruction conjecture is open for the class of connected graphs containing circuits and/or pendant vertices. Such graphs can be expressed uniquely in terms of a trunk, a connected subgraph without pendant vertices, and a tree-growth, a forest each connected component of which meets the trunk in a unique root vertex. This paper establishes the reconstruction conjecture when the trunk is a series-parallel network and the tree-growth is ‘non-trivial’. This is accomplished by means of Tutte's decomposition of the trunk, and group theoretical techniques first developed in [2]. Use is made of the restricted nature of the automorphism groups of the polygon and bond cleavage units of the trunk.