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# 数学代写|图论代考GRAPH THEORY代写|Topological prerequisites

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## 数学代写|图论代写GRAPH THEORY代考|Topological prerequisites

In this section we briefly review some basic topological definitions and facts needed later. All these facts have (by now) easy and well-known proofs; see the notes for sources. Since those proofs contain no graph theory, we do not repeat them here: indeed our aim is to collect precisely those topological facts that we need but do not want to prove. Later, all proofs will follow strictly from the definitions and facts stated here (and be guided by but not rely on geometric intuition), so the material presented now will help to keep elementary topological arguments in those proofs to a minimum.

A straight line segment in the Euclidean plane is a subset of $\mathbb{R}^2$ that has the form ${p+\lambda(q-p) \mid 0 \leqslant \lambda \leqslant 1}$ for distinct points $p, q \in \mathbb{R}^2$. A polygon is a subset of $\mathbb{R}^2$ which is the union of finitely many straight line segments and is homeomorphic to the unit circle $S^1$, the set of points in $\mathbb{R}^2$ at distance 1 from the origin. Here, as later, any subset of a topological space is assumed to carry the subspace topology. A polygonal arc is a subset of $\mathbb{R}^2$ which is the union of finitely many straight line segments and is homeomorphic to the closed unit interval $[0,1]$. The images of 0 and of 1 under such a homeomorphism are the endpoints of this polygonal arc, which links them and runs between them. Instead of ‘polygonal arc’ we shall simply say arc in this chapter. If $P$ is an arc between $x$ and $y$, we denote the point set $P \backslash{x, y}$, the interior of $P$, by $\stackrel{\circ}{P}$.

Let $O \subseteq \mathbb{R}^2$ be an open set. Being linked by an arc in $O$ defines an equivalence relation on $O$. The corresponding equivalence classes are again open; they are the regions of $O$. A closed set $X \subseteq \mathbb{R}^2$ is said to separate $O$ if $O \backslash X$ has more than one region. The frontier of a set $X \subseteq \mathbb{R}^2$ is the set $Y$ of all points $y \in \mathbb{R}^2$ such that every neighbourhood of $y$ meets both $X$ and $\mathbb{R}^2 \backslash X$. Note that if $X$ is open then its frontier lies in $\mathbb{R}^2 \backslash X$.

## 数学代写|图论代写GRAPH THEORY代考|Plane graphs

A plane graph is a pair $(V, E)$ of finite sets with the following properties (the elements of $V$ are again called vertices, those of $E$ edges):
(i) $V \subseteq \mathbb{R}^2$
(ii) every edge is an arc between two vertices;
(iii) different edges have different sets of endpoints;
(iv) the interior of an edge contains no vertex and no point of any other edge.

A plane graph $(V, E)$ defines a graph $G$ on $V$ in a natural way. As long as no confusion can arise, we shall use the name $G$ of this abstract graph also for the plane graph $(V, E)$, or for the point set $V \cup \cup E$; similar notational conventions will be used for abstract versus plane edges, for subgraphs, and so on. ${ }^1$

For every plane graph $G$, the set $\mathbb{R}^2 \backslash G$ is open; its regions are the faces of $G$. Since $G$ is bounded-i.e., lies inside some sufficiently large disc $D$-exactly one of its faces is unbounded: the face that contains $\mathbb{R}^2 \backslash D$. This face is the outer face of $G$; the other faces are its inner faces. We denote the set of faces of $G$ by $F(G)$.

## 数学代写|图论代写GRAPH THEORY代考|Plane graphs

(i) $V \subseteq \mathbb{R}^2$
(ii)每条边都是两个顶点之间的弧;
(iii)不同的边有不同的端点集合;
(iv)一条边的内部不包含任何顶点，也不包含任何其他边的点。

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