如果你也在 怎样代写图论Graph Theory 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。图论Graph Theory有趣的部分原因在于，图可以用来对某些问题中的情况进行建模。这些问题可以在图表的帮助下进行研究(并可能得到解决)。因此，图形模型在本书中经常出现。然而，图论是数学的一个领域，因此涉及数学思想的研究-概念和它们之间的联系。我们选择包含的主题和结果是因为我们认为它们有趣、重要和/或代表主题。
图论Graph Theory通过熟悉许多过去和现在对图论的发展负责的人，可以增强对图论的欣赏。因此，我们收录了一些关于“图论人士”的有趣评论。因为我们相信这些人是图论故事的一部分，所以我们在文中讨论了他们，而不仅仅是作为脚注。我们常常没有认识到数学是一门有生命的学科。图论是人类创造的，是一门仍在不断发展的学科。
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数学代写|图论代写GRAPH THEORY代考|Supply Gas to a Locality
A gas company wants to supply gas to a locality from a single gas source. They are allowed to pass the underground gas lines along the road network only, because no one allows to pass gas lines through the bottom of one’s building. The road network divides the locality into many regions as illustrated in Fig. 1.4(a), where each road is represented by a line segment and a point at which two or more roads meet is represented by a small black circle. A point at which two or more roads meet is called an intersection point. Each region is bounded by some line segments and intersection points. These regions need to be supplied gas. If a gas line reaches an intersection point on the boundary of a region, then the region may receive gas from the line at that intersection point. Thus, the gas lines should reach the boundaries of all the regions of the locality. Gas will be supplied from a gas field which is located outside of the locality, and a single pipe line will be used to supply gas from the gas field to an intersection point on the outer boundary of the locality.
The gas company wants to minimize the establishment cost of gas lines by selecting the roads for laying gas lines such that the total length of the selected roads is minimal. Since gas will be supplied from the gas field using a single line to the locality, the selected road network should be connected and contain an intersection point on the outer boundary of the locality. Thus, the gas company needs to find a set of roads that induces a connected road network, supply gas in all the regions of the locality and the length of the induced road network is minimum. Such a set of roads is illustrated by thick lines in Fig. 1.4(b).The problem mentioned above can be modeled using a “plane graph.” A graph is planar if it can be embedded in the plane without edge crossings. A plane graph is a planar graph with a fixed planar embedding in the plane. A plane graph divides the plane into connected regions called faces. Let $G=(V, E)$ be an edge-weighted connected plane graph, where $V$ and $E$ are the sets of vertices and edges, respectively. Let $F$ be the set of faces of graph $G$. For each edge $e \in E, w(e) \geq 0$ is the weight of the edge $e$ of $G$. A face-spanning subgraph of $G$ is a connected subgraph $H$ induced by a set of edges $S \subseteq E$ such that the vertex set of $H$ contains at least one vertex from the boundary of each face $f \in F$ of $G$ [6]. Figure 1.5 shows two face-spanning subgraphs drawn by thick lines where the cost of the face-spanning subgraph in Fig. 1.5(a) is 22 and the cost of the face-spanning subgraph in Fig. 1.5(b) is 16. Thus, a plane graph may have many face-spanning subgraphs whose costs are different. A minimum face-spanning subgraph $H$ of $G$ is a face-spanning subgraph of $G$, where $\sum_{e \in S} w(e)$ is minimum, and a minimum face-spanning subgraph problem asks to find a minimum face-spanning subgraph of a plane graph. If we represent each road of the road network by an edge of $G$, each intersection point by a vertex of $G$, each region by a face of $G$, and assign the length of a road to the weight of the corresponding edge, then the problem of finding a minimum face-spanning subgraph of $G$ is the same as the problem of the gas company mentioned above [6]. A minimum face-spanning subgraph problem often arises in applications like establishing power transmission lines in a city, power wires layout in a complex circuit, planning irrigation canal networks for irrigation systems, etc.
数学代写|图论代写GRAPH THEORY代考|Floorplanning
Graph modeling has applications in VLSI floorplanning as well as architectural floorplaning [7]. In a VLSI floorplanning problem, an input is a plane graph $F$ as illustrated in Fig. 1.6(a); $F$ represents the functional entities of a chip, called modules, and interconnections among the modules; each vertex of $F$ represents a module, and an edge between two vertices of $F$ represents the interconnections between the two corresponding modules. An output of the problem for the input graph $F$ is a partition of a rectangular chip area into smaller rectangles as illustrated in Fig. 1.6(d); each module is assigned to a smaller rectangle, and furthermore, if two modules have interconnections, then their corresponding rectangles must be adjacent, i.e., they must have a common boundary. A similar problem may arise in architectural floorplanning also. When building a house, the owner may have some preference; for example, a bedroom should be adjacent to a reading room. The owner’s choice of room adjacencies can be easily modeled by a plane graph $F$, as illustrated in Fig. 1.6(a); each vertex represents a room and an edge between two vertices represents the desired adjacency between the corresponding rooms.
A “rectangular drawing” of a plane graph may provide a suitable solution to the floorplanning problem described above. (In a rectangular drawing of a plane graph each vertex is drawn as a point, each edge is drawn as either a horizontal line segment or a vertical line segment and each face including the outer face is drawn as a rectangle.) First, obtain a plane graph $F^{\prime}$ by triangulating all inner faces of $F$ as illustrated in Fig. 1.6(b), where dotted lines indicate new edges added to $F$. Then obtain a “dual-like” graph $G$ of $F^{\prime}$ as illustrated in Fig. 1.6(c), where the four vertices of degree 2 drawn by white circles correspond to the four corners of the rectangular area. Finally, by finding a rectangular drawing of the plane graph $G$, obtain a possible floorplan for $F$ as illustrated in Fig. 1.6(d).
图论代写
数学代写|图论代写GRAPH THEORY代考|Supply Gas to a Locality
某天然气公司希望从单一气源向某一地区供应天然气。他们只允许通过沿着道路网络的地下天然气管道，因为没有人允许通过建筑物底部的天然气管道。如图1.4(a)所示，路网将局地划分为许多区域，其中每条道路用线段表示，两条或两条以上道路相交的点用一个小黑色圆圈表示。两条或两条以上道路相交的点称为交叉点。每个区域由一些线段和交点围合。这些地区需要天然气供应。如果气体管线到达区域边界上的交点，则该区域可以从该交点处的管线接收气体。因此，天然气管道应该到达当地所有地区的边界。天然气由位于局地外的气田供气，由气田供气至局地外边界的交点采用单根管线。
天然气公司希望通过选择铺设天然气管道的道路，使所选道路的总长度最小，从而使天然气管道的建设成本最小化。由于天然气将通过单线从气田输送到该地区，因此所选择的路网应连接并在该地区的外边界上包含一个交叉点。因此，燃气公司需要找到一组道路，该道路可以形成一个连通的路网，在当地的所有地区供气，并且诱导的路网长度最小。这样一组道路如图1.4(b)中的粗线所示。上面提到的问题可以用“平面图”来建模。如果一个图形可以嵌入平面而没有边沿交叉，那么这个图形就是平面的。平面图形是在平面内嵌入固定平面的平面图形。平面图将平面划分为称为面的相连区域。设$G=(V, E)$为边加权连通平面图，其中$V$和$E$分别为顶点和边的集合。设$F$为图形$G$的面集。对于每条边$e \in E, w(e) \geq 0$是$G$的边$e$的权值。$G$的面生成子图是由一组边$S \subseteq E$诱导的连通子图$H$，使得$H$的顶点集至少包含一个来自$G$的每个面$f \in F$的边界的顶点[6]。图1.5显示了用粗线绘制的两个人脸生成子图，其中图1.5(a)中人脸生成子图的代价为22，图1.5(b)中人脸生成子图的代价为16。因此，一个平面图可能有许多代价不同的面生成子图。$G$的最小面部生成子图$H$是$G$的最小面部生成子图，其中$\sum_{e \in S} w(e)$是最小值，最小面部生成子图问题要求找到平面图的最小面部生成子图。如果我们将路网中的每条道路用一条边$G$表示，每个交点用一个顶点$G$表示，每个区域用一个面$G$表示，并将道路的长度分配给相应边的权值，那么寻找$G$最小面生成子图的问题与上面提到的燃气公司问题相同[6]。最小面跨越子图问题经常出现在城市输电线路的建立、复杂电路中的电线布局、灌溉系统灌溉渠网的规划等应用中。
数学代写|图论代写GRAPH THEORY代考|Floorplanning
图形建模在VLSI平面规划和建筑平面规划中都有应用[7]。在VLSI平面规划问题中，输入是如图1.6(a)所示的平面图$F$;$F$表示芯片的功能实体，称为模块，以及模块之间的互连关系;$F$的每个顶点表示一个模块，$F$的两个顶点之间的一条边表示两个对应模块之间的互连。输入图$F$的问题输出是将矩形芯片区域划分为更小的矩形，如图1.6(d)所示;每个模块被分配到一个较小的矩形上，并且，如果两个模块有相互连接，那么它们对应的矩形必须是相邻的，即它们必须有一个共同的边界。在建筑平面规划中也可能出现类似的问题。建房时，业主可能会有一些偏好;例如，卧室应该与阅览室相邻。业主对房间邻接关系的选择可以很容易地用平面图$F$来建模，如图1.6(a)所示;每个顶点表示一个房间，两个顶点之间的边表示相应房间之间所需的邻接关系。
平面图形的“矩形图”可以为上述平面规划问题提供合适的解决方案。(在平面图形的矩形图中，每个顶点绘制为一个点，每个边缘绘制为一个水平线或垂直线，每个面(包括外面)绘制为一个矩形。)首先，如图1.6(b)所示，对$F$的所有内面进行三角剖分，得到一个平面图$F^{\prime}$，虚线表示添加到$F$的新边。然后得到F^{\素数}$的“双象”图$G$，如图1.6(c)所示，其中用白色圆圈画出的4个2度顶点对应于矩形区域的4个角。最后，通过寻找平面图形$G$的矩形图，得到$F$的可能平面图，如图1.6(d)所示。
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