Dung’s abstract theory of argumentation has become established as a general framework for various species of non-monotonic reasoning, and reasoning in the presence of conflict. A Dung framework consists of arguments related by attacks, and the extensions of a framework, and so the status of arguments, are defined under different semantics. Developments of Dung’s work have also defined argument ...

Given a set V of points in the plane, a sequence d 0 ; d 1 ; : : : ; d k of non-negative numbers and an integer n, we are interested in the problem to assign integers from f0; : : : ; n?1g to the points in V such that if x;y 2 V are two points with euclidean distance less than d i , then the diierence between the labels of x and y is not equal to i. This question is inspired by problems occurri...

We present a constructive proof of Ky Fan’s combinatorial lemma concerning labellings of triangulated spheres. Our construction works for triangulations of Sn that contain a flag of hemispheres. As a consequence, we produce a constructive proof of Tucker’s lemma that holds for a larger class of triangulations than previous constructive proofs.

Roman domination in graphs is concerned with the problem of finding a vertex labelling, with minimum weight, satisfaying certain conditions. In this work, the authors initiate the study of a generalization to labellings of both vertices and edges in a graph.

Let G = (V, E) be a finite, simple and undirected graph having v = |V (G)| and e = |E(G)|. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2, . . . , 2q−1} such that, when each edge xy is assigned the label |f(x)−f(y)|, the resulting edge labels are {1, 3, 5, . . . , 2q−1}. Motivated by the work of Z. Gao [6], we have defined odd graceful labeling f...

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i. e. labellings of the nodes by distinct integers of the set {1, . . . , n} in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: recurs...