The context of this paper is the use of formal methods for topology-based geometric modelling. Topology-based geometric modelling deals with objects of various dimensions and shapes. Usually, objects are defined by a graph-based topological data structure and by an embedding that associates each topological element (vertex, edge, face, etc.) with relevant data as their geometric shape (position...

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.

The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in a planar graph of size n. Two naïve solutions are to store the graph using Opnq bits, or to explicitly store the distance matrix with Opk log nq bits. The only lower bounds are from the seminal work of Gavoille, Peleg, Prennes, and Raz [SODA’01], who rule out compressions into a polynomially smalle...

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In the paper we establish some bounds for the number of edges in supermagic graphs.

The distance-two labeling problem of graphs was proposed by Griggs and Roberts in 1988, and it is a variation of the frequency assignment problem introduced by Hale in 1980. An L(2, 1)-labeling of a graph G is an assignment of nonnegative integers to the vertices of G such that vertices at distance two receive different numbers and adjacent vertices receive different and nonconsecutive integers...

In all the paper we work on a graphG with maximal degree∆. For a set of vertices S ⊂ V (G), the graph G\S is the graph induced by V (G)\S. The distance d(u, v) between two vertices u and v is the number of edges in the shortest path from u to v. We say that v is a d-neighbor of u if d(u, v) = d. Let Nd(v) be the set of d-neighbors of v. We will generally use the common term neighbor instead of ...

let $g$ be an $(n,m)$-graph. we say that $g$ has property $(ast)$if for every pair of its adjacent vertices $x$ and $y$, thereexists a vertex $z$, such that $z$ is not adjacentto either $x$ or $y$. if the graph $g$ has property $(ast)$, thenits complement $overline g$ is connected, has diameter 2, and itswiener index is equal to $binom{n}{2}+m$, i.e., the wiener indexis insensitive of any other...

two digraphs of same order are said to be skew-equienergetic if theirskew energies are equal. one of the open problems proposed by li andlian was to construct non-cospectral skew-equienergetic digraphs onn vertices. recently this problem was solved by ramane et al. inthis paper, we give some new methods to construct new skew-equienergeticdigraphs.

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.