A univariate polynomial f over a field is decomposable if it is the composition f = g ◦h of two polynomials g and h whose degree is at least 2. We determine an approximation to the number of decomposable polynomials over a finite field. The tame case, where the field characteristic p does not divide the degree n of f , is reasonably well understood, and we obtain exponentially decreasing error ...

A univariate polynomial f over a field is decomposable if it is the composition f = g ◦h of two polynomials g and h whose degree is at least 2. We determine an approximation to the number of decomposable polynomials over a finite field. The tame case, where the field characteristic p does not divide the degree n of f , is reasonably well understood, and we obtain exponentially decreasing error ...

Let K denote a field. A polynomial f(x) ∈ K[x] is said to be decomposable over K if f(x)= g(h(x)) for some polynomials g(x) and h(x)∈K[x] with 1< deg(h) < deg(f ). Otherwise f(x) is called indecomposable. If f(x)= g(xm) with m> 1, then f(x) is said to be trivially decomposable. In this paper, we show that xd+ax+b is indecomposable and that if e denotes the largest proper divisor of d, then xd+a...

Given a chordal graph G, and a class of subgraphs H of G, we investigate the problem of finding an element H ∈ H which minimizes a function f : H → R. We show that when f satisfies a decomposability criterion, and when H admits what we call an efficiently decomposable configuration space with respect to f , then this optimization problem can be solved in polynomial time. We give a dynamic progr...

This thesis consists of four papers studying structure learning and Bayesian inference in probabilistic graphical models for both undirected and directed acyclic graphs (DAGs). Paper A presents a novel algorithm, called the Christmas tree algorithm (CTA), that incrementally construct junction trees for decomposable graphs by adding one node at a time to the underlying graph. We prove that CTA w...

A graph G is locally irregular if every two adjacent vertices of G have different degrees. A locally irregular decomposition of G is a partition E1, ..., Ek of E(G) such that each G[Ei] is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail, Przyby lo and Woźniak that they decompose i...

A walk on an edge-colored graph G is a path in G which contains each edge at least once, and the trace of a walk w is the sequence of edge-colors seen in w. We investigate the walk realizability problem, which is to decide, given an edge-colored graph G and a string x of colors, if there is a walk on G with trace x. We de ne a method of decomposing graphs into linear chains, and show that the n...

He, Hou, Lih, Shao, Wang, and Zhu showed that a planar graph of girth 11 can be decomposed into a forest and a matching. Borodin, Kostochka, Sheikh, and Yu improved the bound on girth to 9. We give sufficient conditions for a planar graph with 3-cycles to be decomposable into a forest and a matching. c © 2008 Elsevier B.V. All rights reserved.

The proof of the following theorem is the main result of this paper: If G is a 2k-regular graph that has a perfect 1-factorisation, then the line graph, L(G), of G is Hamilton decomposable. Consideration is given to Hamilton decompositions of L(K 2k ? F).