This paper surveys the applications of Smarandache’s notion to graph theory appeared in International J.Math.Combin. from Vol.1,2008 to Vol.3,2009. In fact, many problems discussed in these papers are generalized in this paper. Topics covered in this paper include: (1)What is a Smarandache System? (2)Vertex-Edge Labeled Graphs with Applications: (i)Smarandachely k-constrained labeling of a grap...

The EU-funded project Lynx focuses on the creation of a knowledge graph for legal domain (Legal Knowledge Graph, LKG) and its use semantic processing, analysis enrichment documents from domain. This article describes cases covered in project, entire developed platform services that operate documents.

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space, denoted WCW (G). Let B be a complete bipartite induced subgraph of ...

We give a very simple proof that every non-bipartite matching covered graph contains a nice subgraph that is an odd subdivision of K 4 or C 6. It follows immediately that every brick diierent from K 4 and C 6 has an edge whose removal preserves the matching covered property. These are classical and very useful results due to Lovv asz.

Let G = (V (G),E(G)) be a graph. and let g, f be two integer-valued functions defined on V (G) such that g(x)≤ f (x) for all x ∈V (G). G is called fractional (g, f )-covered if each edge e of G belongs to a fractional (g, f )-factor Gh such that h(e) = 1, where h is the indicator function of Gh. In this paper, sufficient conditions related to toughness and isolated toughness for a graph to be f...

A graph is Zm-well-covered if all maximal independent sets have the same cardinality modulo m. Zm-well-covered graphs generalise well-covered graphs, those in which all independent sets have the same cardinality. Z2-well-covered graphs are also called parity graphs. A characterisation of cubic well-covered graphs was given by Campbell, Ellingham and Royle. Here we extend this to a characterisat...

─ Graph theory is atool for solving Capacitated Vehicle Routing Problem (CVRP). This paper describes the application of software based on graph theory to analyze and solve CVRP in a nationwide freight company.The analysis was performed on a fleet of 13 vehicles with similar technical specifications and load capacity (15 tons). Medellin and its metropolitan area municipalities were the origin ch...

This paper examines the competitiveness of winner-take-all price competition in homogeneous product oligopoly environments where underlying buyer demands and/or firms’ costs need not be continuous. Our analysis is motivated by the observation that a variety of economic settings have these features. For example, in 1996 an Ivy League university solicited bids from several vendors for its initiat...

A bipartite graph is said to be symmetric if it has symmetry of reflecting two vertex sets. This paper investigates matching structure of symmetric bipartite graphs. We first apply the Dulmage-Mendelsohn decomposition to a symmetric bipartite graph. The resulting components, which are matching-covered, turn out to have symmetry. We then decompose a matching-covered bipartite graph via an ear de...

A weighted graph G is called well-covered if all its maximal independent sets have the same weight. Let S be an independent set of G (possibly, S = ∅). The subgraph G − N [S] is called a co-stable subgraph of G. We denote by CSub(G) the set of all co-stable subgraphs of G considered up to isomorphism. A class of weighted graphs P is called co-hereditary if it is closed under taking co-stable su...