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# 数学代写|离散数学代写Discrete Mathematics代考|MATH215 Planar Graphs

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## 数学代写|离散数学代写Discrete Mathematics代考|Planar Graphs

Suppose we have a graph $G$ and that we want to draw it “nicely” on a piece of paper, which means that we draw the vertices as points and the edges as line segments joining some of these points, in such a way that no two edges cross each other, except possibly at common endpoints. We have more flexibility and still have a nice picture if we allow each abstract edge to be represented by a continuous simple curve (a curve that has no self-intersection), that is, a subset of the plane homeomorphic to the closed interval $[0,1]$ (in the case of a loop, a subset homeomorphic to the circle, $S^1$ ). If a graph can be drawn in such a fashion, it is called a planar graph. For example, consider the graph depicted in Figure 10.54.

If we look at Figure 10.54, we may believe that the graph $G$ is not planar, but this is not so. In fact, by moving the vertices in the plane and by continuously deforming some of the edges, we can obtain a planar drawing of the same graph, as shown in Figure $10.55$.

However, we should not be overly optimistic. Indeed, if we add an edge from node 5 to node 4 , obtaining the graph known as $K_5$ shown in Figure 10.56, it can be shown that there is no way to move the nodes around and deform the edge continuously to obtain a planar graph (we prove this a little later using the Euler formula). Another graph that is nonplanar is the bipartite grapk $K_{3,3}$. The two graphs, $K_5$ and $K_{3,3}$ play a special role with respect to planarity. Indeed, a famous theorem of $\mathrm{Ku}-$ ratowski says that a graph is planar if and only if it does not contain $K_5$ or $K_{3,3}$ as a minor (we explain later what a minor is).

## 数学代写|离散数学代写Discrete Mathematics代考|Criteria for Planarity

Let us now go back to Kuratowski’s criterion for nonplanarity. For this it is useful to introduce the notion of edge contraction in a graph.

Definition 10.32. Let $G=(V, E, s t)$ be any graph and let $e$ be any edge of $G$. The graph obtained by contracting the edge e into a new vertex $v_e$ is the graph $G / e=$ $\left(V^{\prime}, E^{\prime}, s t^{\prime}\right)$ with $V^{\prime}=(V-s t(e)) \cup\left{v_e\right}$, where $v_e$ is a new node $\left(v_e \notin V\right) ; E^{\prime}=$ $E-{e} ;$ and with
$$s t^{\prime}\left(e^{\prime}\right)= \begin{cases}s t\left(e^{\prime}\right) & \text { if } s t\left(e^{\prime}\right) \cap s t(e)=\emptyset \ \left{v_e\right} & \text { if } s t\left(e^{\prime}\right)=s t(e) \ \left{u, v_e\right} & \text { if } s t\left(e^{\prime}\right) \cap s t(e)={z} \text { and } s t\left(e^{\prime}\right)={u, z} \text { with } u \neq z \ \left{v_e\right} & \text { if } s t\left(e^{\prime}\right)={x} \text { or } s t\left(e^{\prime}\right)={y} \text { with } s t(e)={x, y} .\end{cases}$$
If $G$ is not a simple graph, then we need to eliminate parallel edges and loops. In this case, $e={x, y}$ and $G / e=\left(V^{\prime}, E^{\prime}, s t\right)$ is defined so that $V^{\prime}=(V-{x, y}) \cup\left{v_e\right}$, where $v_e$ is a new node and
\begin{aligned} E^{\prime}=&{{u, v} \mid{u, v} \cap{x, y}=\emptyset} \ & \cup\left{\left{u, v_e\right} \mid{u, x} \in E-{e} \quad \text { or } \quad{u, y} \in E-{e}\right} . \end{aligned}
Figure $10.61$ shows the result of contracting the upper edge ${2,4}$ (shown as a thicker line) in the graph shown on the left, which is not a simple graph.

## 数学代写|离散数学代写离散数学代考|平面性标准

$$s t^{\prime}\left(e^{\prime}\right)= \begin{cases}s t\left(e^{\prime}\right) & \text { if } s t\left(e^{\prime}\right) \cap s t(e)=\emptyset \ \left{v_e\right} & \text { if } s t\left(e^{\prime}\right)=s t(e) \ \left{u, v_e\right} & \text { if } s t\left(e^{\prime}\right) \cap s t(e)={z} \text { and } s t\left(e^{\prime}\right)={u, z} \text { with } u \neq z \ \left{v_e\right} & \text { if } s t\left(e^{\prime}\right)={x} \text { or } s t\left(e^{\prime}\right)={y} \text { with } s t(e)={x, y} .\end{cases}$$
。如果$G$不是一个简单的图，那么我们需要消除平行边和循环。在本例中，$e={x, y}$和$G / e=\left(V^{\prime}, E^{\prime}, s t\right)$被定义为$V^{\prime}=(V-{x, y}) \cup\left{v_e\right}$，其中$v_e$是一个新节点，而
\begin{aligned} E^{\prime}=&{{u, v} \mid{u, v} \cap{x, y}=\emptyset} \ & \cup\left{\left{u, v_e\right} \mid{u, x} \in E-{e} \quad \text { or } \quad{u, y} \in E-{e}\right} . \end{aligned}

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。