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Euler's Formula is a cornerstone of graph theory and plays a critical role in understanding planar graphs. It provides a simple, yet powerful relationship between the number of vertices (V), edges (E), and faces (F) in a connected planar graph. The formula states that for any connected planar graph, \[ V - E + F = 2 \] This equation is pivotal when determining the possibility of a graph's planarity, as was considered in the example of a graph with 10 vertices and edges. By applying Euler's Formula, we can deduce the maximum number of edges a graph with a given number of vertices can have and still be planar. This relationship is not only fundamental in graph theory but also in the fields of computer science, biology, and civil engineering, among others, where planar graphs model networks free from intersecting connections.

When discussing graphs in mathematics, a connected graph is one where each pair of vertices is linked by a path. In the context of planar graphs, a connected graph is one that remains intact, meaning there's always a way to travel from one vertex to any other without lifting your pencil from the plane.

Graph theory is a flourishing field of mathematics focused on the study of graphs, which are mathematical structures consisting of vertices (also called nodes) and edges (which connect the vertices). Although it might sound abstract, graph theory is incredibly applicable in our daily lives, from organizing efficient delivery routes to modeling the spread of diseases in epidemiology. It provides the language and tools needed to analyze various types of networks, whether they're social, informational, or biological. The problem presented showcases a practical application within graph theory, examining the conditions under which a particular type of graph, a planar graph, can exist.

In graph theory, the fundamental components are vertices and edges. Vertices (or nodes) can be thought of as points, while edges represent the connections between these points. In any graph, the way these vertices and edges are arranged can widely vary, leading to different types of graphs with distinct properties. Planar graphs, in specific, can help us visually understand many real-world systems, as they must be drawn such that no edges cross. This inherent characteristic of planar graphs limits the number of edges that can be present, which is at the heart of determining a graph's planarity—as illustrated in the given exercise.