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# 数学代写|图论代考GRAPH THEORY代写|The structure of 3-connected graphs

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## 数学代写|图论代写GRAPH THEORY代考|The structure of 3-connected graphs

In the last section we showed first how every connected graph decomposes canonically into 2 -connected subgraphs (and bridges), and how these are arranged in a tree-like way to make up the whole graph. There is a similar canonical decomposition of 2-connected graphs into 3-connected pieces (and cycles), which are again organized in a tree-like way. This nontrivial structure theorem of Tutte is most naturally expressed in terms of tree-decompositions, to be introduced in Chapter 12. We therefore omit it here. $^1$

Instead, we shall describe how every 3 -connected graph can be obtained from a $K^4$ by a succession of elementary operations preserving 3 -connectedness. We then prove a deep result of Tutte about the algebraic structure of the cycle space of 3-connected graphs; this will play an important role again in Chapter 4.5.

In Proposition 3.1.3 we saw how every 2-connected graph can be constructed inductively by a sequence of steps starting from a cycle. All the graphs in the sequence were themselves 2-connected, so the graphs obtainable by this construction method are precisely the 2-connected graphs. Note that the cycles as starting graphs cannot be replaced by a smaller class, because they do not have proper 2-connected subgraphs.
When we try to do the same for 3-connected graphs, we soon notice that both the set of starting graphs and the construction steps required become too complicated. If we base our construction sequences on the minor relation instead of subgraphs, however, it all works smoothly again:

Lemma 3.2.1. If $G$ is 3-connected and $|G|>4$, then $G$ has an edge $e$ such that $G / e$ is again 3-connected.

Proof. Suppose there is no such edge $e$. Then, for every edge $x y \in G$, the graph $G / x y$ contains a separator $S$ of at most 2 vertices. Since $\kappa(G) \geqslant 3$, the contracted vertex $v_{x y}$ of $G / x y$ (see Chapter 1.7) lies in $S$ and $|S|=2$, i.e. $G$ has a vertex $z \notin{x, y}$ such that $\left{v_{x y}, z\right}$ separates $G / x y$. Then any two vertices separated by $\left{v_{x y}, z\right}$ in $G / x y$ are separated in $G$ by $T:={x, y, z}$. Since no proper subset of $T$ separates $G$, every vertex in $T$ has a neighbour in every component $C$ of $G-T$.

## 数学代写|图论代写GRAPH THEORY代考|Menger’s theorem

The following theorem is one of the cornerstones of graph theory.
Theorem 3.3.1. (Menger 1927)
Let $G=(V, E)$ be a graph and $A, B \subseteq V$. Then the minimum number of vertices separating $A$ from $B$ in $G$ is equal to the maximum number of disjoint $A-B$ paths in $G$.

We offer three proofs. Whenever $G, A, B$ are given as in the theorem, we denote by $k=k(G, A, B)$ the minimum number of vertices separating $A$ from $B$ in $G$. Clearly, $G$ cannot contain more than $k$ disjoint $A-B$ paths; our task will be to show that $k$ such paths exist.

First proof. We apply induction on $|G|$. If $G$ has no edge, then $|A \cap B|=k$ and we have $k$ trivial $A-B$ paths. So we assume that $G$ has an edge $e=x y$. If $G$ has no $k$ disjoint $A-B$ paths, then neither does $G / e$; here, we count the contracted vertex $v_e$ as an element of $A$ (resp. $B$ ) in $G / e$ if in $G$ at least one of $x, y$ lies in $A$ (resp. $B$ ). By the induction hypothesis, $G / e$ contains an $A-B$ separator $Y$ of fewer than $k$ vertices. Among these must be the vertex $v_e$, since otherwise $Y \subseteq V$ would be an $A-B$ separator in $G$. Then $X:=\left(Y \backslash\left{v_e\right}\right) \cup{x, y}$ is an $A-B$ separator in $G$ of exactly $k$ vertices.

We now consider the graph $G-e$. Since $x, y \in X$, every $A-X$ separator in $G-e$ is also an $A-B$ separator in $G$ and hence contains at least $k$ vertices. So by induction there are $k$ disjoint $A-X$ paths in $G-e$, and similarly there are $k$ disjoint $X-B$ paths in $G-e$. As $X$ separates $A$ from $B$, these two path systems do not meet outside $X$, and can thus be combined to $k$ disjoint $A-B$ paths.

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