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# 数学代写|图论代考GRAPH THEORY代写|Math781 Connectivity and Paths

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## 数学代写|图论代写GRAPH THEORY代考|Connectivity and Paths

Now that we have some familiarity with connectivity, we turn to its relationship to paths within a graph. Note that for the remainder of this section, we will assume the graphs are connected, as otherwise the results are trivial. We begin by relating cut-vertices and bridges to paths. You should notice that almost every result for vertices has an edge analog. We begin with the most simple results relating a cut-vertex or a bridge to its presence on a path.
Theorem 4.6 A vertex $v$ is a cut-vertex of a graph $G$ if and only if there exist vertices $x$ and $y$ such that $v$ is on every $x-y$ path.
Proof: First suppose $v$ is a cut-vertex in a graph $G$. Then $G-v$ must have at least two components. Let $x$ and $y$ be vertices in different components of $G-v$. Since $G$ is connected, we know there must exist an $x-y$ path in $G$ that does not exist in $G-v$. Thus $v$ must lie on this path.

Conversely, let $v$ be a vertex and suppose there exist vertices $x$ and $y$ such that $v$ is on every $x-y$ path. Then none of these paths exist in $G-v$, and so $x$ and $y$ cannot be in the same component of $G-v$. Thus $G$ must have at least two components and so $v$ is a cut-vertex.

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

The following theorems generalize the results above relating a cut-vertex or bridge to paths in a graph. Menger’s Theorem, and the resulting theorems, show the number of internally disjoint (or edge-disjoint) paths directly corresponds to the connectivity (or edge-connectivity) of a graph. For example, in $G_2$ above we could separate $b$ and $c$ using two vertices and it should be easy to see that $b h c$ and $b e f d c$ are internally disjoint $b-c$ paths. However, if we try to find more than two $b-c$ paths then one of them cannot be internally disjoint from the others (try it!).

Theorem 4.13 (and its edge analog) is named for Karl Menger, the Austrian-American mathematician who first published the result in 1927 [65]. There are many different versions of the proof, and the one presented here

most closely resembles that in [21]. Note that for this proof we need an additional process, call a contraction.

Definition 4.12 Let $e=x y$ be an edge of a graph $G$. The contraction of $e$, denoted $G / e$, replaces the edge $e$ with a vertex $v_e$ so that any vertices adjacent to either $x$ or $y$ are now adjacent to $v_e$. Contracting an edge creates a smaller graph, both in terms of the number of vertices and edges, but keeps much of the structure of a graph in tact. In particular, contracting an edge cannot disconnect a graph (see Exercise 4.22).

## MATLAB代写

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