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# 数学代写|图论代考GRAPH THEORY代写|MATH3020

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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).