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When studying planar graphs, one of the fundamental concepts you'll encounter is Kuratowski's theorem. It provides a crucial test for checking whether a given graph can be drawn on a plane without any edges crossing each other. According to this theorem, a graph is planar if and only if it does not contain a subgraph that is a subdivision of either the complete graph on five vertices, denoted as \( \mathrm{K}_{5} \), or the complete bipartite graph on six vertices, denoted as \( \mathrm{K}_{3,3} \).

A subdivision occurs when you add new vertices along the edges of these graphs, but the intrinsic connectivity doesn't change. It tells us that if we're trying to unravel a complex drawing to see if it's a planar graph, we're looking for these two specific configurations as warning signs. They are the deal-breakers for planarity. In our exercise, the presence of \( \mathrm{K}_{5} \) itself within the graph automatically means it violates this theorem and therefore, isn't planar.

What is a Complete Graph?
In graph theory, a complete graph is like a social network where everyone knows everyone else; it's a set of vertices with an edge between every pair of vertices. The number of edges in a complete graph with \( n \) vertices is given by the simple formula \( \binom{n}{2} \), which represents choosing any two vertices to draw an edge.

The complete graph with \( n \) vertices is denoted by \( \mathrm{K}_{n} \). One important property of complete graphs is that they are highly connected, meaning there's an abundance of paths to move from one vertex to another. This high connectivity also means that complete graphs are not planar when \( n > 4 \), as their complex interconnections inevitably require edges to cross when drawn on a plane.

Understanding the \( \mathrm{K}_{5} \) Graph
Let's zoom in on the \( \mathrm{K}_{5} \) graph, which stands as the simplest non-planar complete graph. The \( \mathrm{K}_{5} \) graph has five vertices and every vertex is connected to every other vertex, for a total of \( \binom{5}{2} = 10 \) edges. It's a network of interconnecting lines that looks somewhat like a star with additional connections.

This richness of connections makes it impossible to draw \( \mathrm{K}_{5} \) on a flat surface without some of the edges crossing. Here lies the reason for its non-planarity: as we add more vertices, the complexity of the graph increases, and with \( \mathrm{K}_{5} \) we hit the threshold where planarity is lost. This serves as a solid example for understanding the broader implications of Kuratowski's theorem and why, in our exercise, \( \mathrm{K}_{5} \) cannot be considered a planar graph.