O menor limite inferior de vértices de grau 2 para um grafo minimal 2-aresta-conexo

Abstract

Studying graph theory is undoubtedly an experience that shows us the importance of this subject due to its applicability in several branches of Mathematics as well as its notorious appearance in everyday situations. In this sense, our work aims at first to base the reader on the initial concepts of graphs such as their elements (vertices and edges), valence of each vertex, types of walk, isomorphism and operations with graphs. From this, we present the concept of connectedness so that the reader can recognize a connected or disconnected graph, the definitions of trees and forests as well as their properties. Familiar with these concepts, we then insert the concept of minimal 2 − edge − connected graphs. Using the rigor of mathematics, we test by induction the equation that brings us to the lowest lower bound of vertices of degree 2 in a minimal 2 − edge − connected graph. At this point, we are faced with an inconsistency. We found a case where the minimality of the graph was not observed but, however, the effectiveness of the equation was not compromised. After that, for the modeling of graphs, we use programs and software such as geogebra, google maps and Lucidchart, the latter specific for the assembly of network systems. We also cite the use of concrete materials such as the Geoplane that consists of a platform with pins and alloys, completely manipulable that were used in the application of a playful activity with 32 students of high school / technical in logistics which helped us in the didactic question, enabling a greater interaction between the people involved in the construction process. The proposal to involve them in this activity of obtaining a viable connection system applicable in their logistics projects generated interesting results and questions that aroused their desire to delve deeper into the theme.