# 数学代写|图论作业代写Graph Theory代考|Algebraic planarity criteria

#### Doug I. Jones

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## 数学代写|图论作业代写Graph Theory代考|Algebraic planarity criteria

One of the most conspicuous features of a plane graph $G$ are its facial cycles, the cycles that bound a face. If $G$ is 2 -connected it is covered by its facial cycles, so in a sense these form a ‘large’ set. In fact, the set of facial cycles is large even in the sense that they generate the entire cycle space: every cycle in $G$ is easily seen to be the sum of the facial cycles (see below). On the other hand, the facial cycles only cover $G$ ‘thinly’, as every edge lies on at most two of them. Our first aim in this section is to show that the existence of such a large yet thinly spread family of cycles is not only a conspicuous feature of planarity but lies at its very heart: it characterizes it.

Let $G=(V, E)$ be any graph. We call a subset $\mathcal{F}$ of its edge space $\mathcal{E}(G)$ simple if every edge of $G$ lies in at most two sets of $\mathcal{F}$. For example, the cut space $\mathcal{C}^*(G)$ has a simple basis: according to Proposition 1.9.3 it is generated by the cuts $E(v)$ formed by all the edges at a given vertex $v$, and an edge $x y \in G$ lies in $E(v)$ only for $v=x$ and for $v=y$.
Theorem 4.5.1. (MacLane 1937)
A graph is planar if and only if its cycle space has a simple basis.
Proof. The assertion being trivial for graphs of order at most 2, we consider a graph $G$ of order at least 3. If $\kappa(G) \leqslant 1$, then $G$ is the union of two proper induced subgraphs $G_1, G_2$ with $\left|G_1 \cap G_2\right| \leqslant 1$. Then $\mathcal{C}(G)$ is the direct sum of $\mathcal{C}\left(G_1\right)$ and $\mathcal{C}\left(G_2\right)$, and hence has a simple basis if and only if both $\mathcal{C}\left(G_1\right)$ and $\mathcal{C}\left(G_2\right)$ do (proof?). Moreover, $G$ is planar if and only if both $G_1$ and $G_2$ are: this follows at once from Kuratowski’s theorem, but also from easy geometrical considerations. The assertion for $G$ thus follows inductively from those for $G_1$ and $G_2$. For the rest of the proof, we now assume that $G$ is 2-connected.

## 数学代写|图论作业代写Graph Theory代考|Plane duality

In this section we shall use MacLane’s theorem to uncover another connection between planarity and algebraic structure: a connection between the duality of plane graphs, defined below, and the duality of the cycle and cut space hinted at in Chapters 1.9 and 2.4.

A plane multigraph is a pair $G=(V, E)$ of finite sets (of vertices and edges, respectively) satisfying the following conditions:
(i) $V \subseteq \mathbb{R}^2$;
(ii) every edge is either an arc between two vertices or a polygon containing exactly one vertex (its endpoint);
(iii) apart from its own endpoint(s), an edge contains no vertex and no point of any other edge.

We shall use terms defined for plane graphs freely for plane multigraphs. Note that, as in abstract multigraphs, both loops and double edges count as cycles.

Let us consider the plane multigraph $G$ shown in Figure 4.6.1. Let us place a new vertex inside each face of $G$ and link these new vertices up to form another plane multigraph $G^$, as follows: for every edge $e$ of $G$ we link the two new vertices in the faces incident with $e$ by an edge $e^$ crossing $e$; if $e$ is incident with only one face, we attach a loop $e^$ to the new vertex in that face, again crossing the edge $e$. The plane multigraph $G^$ formed in this way is then dual to $G$ in the following sense: if we apply the same procedure as above to $G^$, we obtain a plane multigraph very similar to $G$; in fact, $G$ itself may be reobtained from $G^$ in this way.

# 图论代考

## 数学代写|图论作业代写Graph Theory代考|Plane duality

(i) $V \subseteq \mathbb{R}^2$;
(ii)每条边要么是两个顶点之间的弧，要么是一个只包含一个顶点(其端点)的多边形;
(iii)一条边除了自己的端点外，不包含任何顶点，也不包含任何其他边的点。

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