Assume that G = (V,E) is an undirected graph, and C ⊆ V . For every v ∈ V , we denote by I(v) the set of all elements of C that are within distance one from v. If the sets I(v) \ {v} for v ∈ V are all nonempty, and, moreover, the sets {I(v), I(v) \ {v}} for v ∈ V are disjoint, then C is called a strongly identifying code. The smallest possible density of a strongly identifying code in the infin...

Let $G$ be a simple connected graph. In this paper, Szeged dimension and PI$_v$ dimension of graph $G$ are introduced. It is proved that if $G$ is a graph of Szeged dimension $1$ then line graph of $G$ is 2-connected. The dimensions of five composite graphs: sum, corona, composition, disjunction and symmetric difference with strongly regular components is computed. Also explicit formulas of Sze...

Tutte introduced the theory of nowhere zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere zero A-flow, for any Abelian group A with |A| ≥ k. In 1992, Jaeger et al. [9] extended nowhere zero flows to group connectivity of graphs: given an orientationD of a graph G, if for any b : V (G) → Awith  v∈V (G) b(v) = 0, there always exists a map f : E(G) →...

Linguists often represent the relationships between words in a collection of text as an undirected graph G = (V,E), were V is the vocabulary and vertices are adjacent in G if and only if the words they represent co-occur in a relevant pattern in the text. Ideally, the words with similar meanings give rise to the vertices of a component of the graph. However, many words have several distinct mea...

A graph G = (V,E) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if (x, y) ∈ E. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A face subdivision of a triangular grid graph is replacing some of its cells by plane copies of the...

Maximal outerplanar graphs are characterized using three different classes of graphs. A path-neighborhood graph is a connected graph in which every neighborhood induces a path. The triangle graph T (G) has the triangles of the graph G as its vertices, two of these being adjacent whenever as triangles in G they share an edge. A graph is edge-triangular if every edge is in at least one triangle. ...

We present some algorithms which construct a triangular graph given a set of points, where each face of the graph complies with the Delaunay criteria. Then we develop an linear time algorithm to construct the contour of such a triangular graph, where each face is coloured either black or white. Our techniques avoid expensive trigonometric computations. We intoduce our algorithms for the 2-dimen...