Regular coverings of a graph or a map admit descriptions in terms of certain group-valued labellings on the graph. Controlling the properties of graphs resulting from such coverings requires to develop a certain type of ‘labelling calculus’. We give a brief outline of the related theory and survey the most important results.

A graph G is a k-dot product graph if there exists a vector labelling u : V (G)→ Rk such that u(i)Tu(j) ≥ 1 if and only if ij ∈ E(G). Fiduccia, Scheinerman, Trenk and Zito [4] asked whether every planar graph is a 3-dot product graph. We show that the answer is “no”. On the other hand, every planar graph is a 4-dot product graph. We also answer the corresponding questions for planar graphs of p...

In order to apply canonical labelling of graphs and isomorphism checking in interactive theorem provers, these algorithms must either be mechanically verified or their results verifiable by independent checkers. We analyze a state-of-the-art algorithm for (described McKay Piperno) formulate it terms formal proof system. provide an implementation that can export the obtained graph is form given ...

It is the powering construction which will be of interest to us. Essentially, the powering construction works by taking a constraint graph, and building a t-fold power graph, which collects all paths of length t emanating from a node. The proof (i.e. the labelling of the constraint graph) now must supply versions of labels of nodes collected at each composite node – of course, the constraints b...

Let σ, τ : N→N be two sequences of natural numbers. We say that σ dominates τ if σ(n) τ(n) for every n greater than some n0 ∈ N. Now let G be an infinite graph, and assume the vertices of G are labelled with natural numbers, say by f : V (G)→N. Then each ray (one-way infinite path) R = x0x1 . . . in G gives rise to a sequence τR := (f(x0), f(x1), . . .); we say that σ dominates R if σ dominates...

This paper formulates pixel labelling as a series of two-category classification. Unlike existing techniques, which assign a determinate label to each pixel, we assign a label set to each pixel and shrink the label set step by step. Determinate labelling is achieved within log2n (n is size of label set) steps. In each step, we bisect the label set into two subsets and discard the one with highe...

An L(p, 1)-labelling of a graph is a function f from the vertex set to the positive integers such that |f(x) − f(y)| ≥ p if dist(x, y) = 1 and |f(x)− f(y)| ≥ 1 if dist(x, y) = 2, where dist(x, y) is the distance between the two vertices x and y in the graph. The span of an L(p, 1)labelling f is the difference between the largest and the smallest labels used by f plus 1. In 1992, Griggs and Yeh ...

We introduce a complexity measure of modal properties of finite structures which generalises the automaticity of languages. It is based on graph-automata like devices called labelling systems. We define a measure of the size of a structure that we call rank, and show that any modal property of structures can be approximated up to any fixed rank n by a labelling system. The function that takes n...

In pattern recognition applications, it is useful to represent objects by attributed graphs considering their structural properties. Besides, some graph matching problems need a Common Labelling between vertices of a set of graphs. Computing this Common Labelling is an NP-complete problem. Stateof-the-art algorithms are composed by two steps: in the first, they compute all pairwise labellings a...