Let G = (V, E) be a graph or digraph and r : V → Z+. An r-detachment of G is a graph H obtained by ‘splitting’ each vertex v ∈ V into r(v) vertices. The vertices v1, ..., vr(v) obtained by splitting v are called the pieces of v in H. Every edge uv ∈ E corresponds to an edge of H connecting some piece of u to some piece of v. Crispin Nash-Williams [9] gave necessary and sufficient conditions for...

Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals ...

Directed Acyclic Graphs are often used to model circuits and networks. The path length in such Directed Acyclic Graphs represents circuit or network delays. In the vertex splitting problem, the objective is to determine a minimum number of vertices from the graph to split such that the resulting graph has no path of length greater than a given δ. The problem has been proven to be NP-hard. A Gen...

We present new complexity results on the compilation of CNFs into DNNFs and OBDDs. In particular, we introduce a new notion of width, called CV-width, which is specific to CNFs and that dominates the treewidth of the CNF incidence graph. We then show that CNFs can be compiled into structured DNNFs in time and space that are exponential only in CV-width. Not only does CV-width dominate the incid...

Moriah and Schultens have demonstrated that an irreducible Heegaard splitting of an orientable Seifert fibered space over an orientable base surface is either vertical or horizontal. In this paper it is determined precisely which vertical and horizontal splittings are irreducible. Let M be a Seifert fibered space which admits a horizontal splitting at the fiber f . If the genus of the horizonta...

it is necessary to generate the automorphism group of a chemical graph in computer-aidedstructure elucidation. an euclidean graph associated with a molecule is defined by a weightedgraph with adjacency matrix m = [dij], where for i≠j, dij is the euclidean distance between thenuclei i and j. in this matrix dii can be taken as zero if all the nuclei are equivalent. otherwise,one may introduce dif...

a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are disti...

The goal of this work is to derive and justify a multilevel preconditioner of optimal arithmetic complexity for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. Our approach is based on the following simple idea given in [R.D. Lazarov, P.S. Vassilevski, L.T. Zikatanov, Multilevel preconditioning of second order elliptic discontin...

Effects of ATP and Mg2+ concentrations on bioluminescence reaction of luciferase (Photinus pyralis) were investigated by home-made luminometer. The Michaelis constants of the enzyme for ATP and Mg2+ obtained from the Lineweaver-Burk graph, were 61.9 mM ±3.3 mM and 251.6mM ± 39.0mM...