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# 数学代写|图论代写graph theory代考|2-Connected Graphs

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## 数学代写|图论代写graph theory代考|2-Connected Graphs

As we have already seen, 1-connected graphs are simply those graphs that we more commonly call connected and $k$-connected graphs can be described in terms of $k$ number of paths between two vertices. So why then do we single out 2-connected graphs? This is in part because they hold a special area in the study of connectivity – they are known to be connected and as we will see cannot contain any cut-vertices. The class of 2-connected graphs provide both some easy results and some more technical and complex areas of study. We begin with a review of our graphs $G_{2}$ and $G_{3}$ from page 169 , reproduced below.

Recall that we showed $\kappa\left(G_{2}\right)=2=\kappa^{\prime}\left(G_{2}\right)$ but that $\kappa\left(G_{3}\right)=1$ and $\kappa^{\prime}\left(G_{3}\right)=2$. So what is the structural difference between $G_{2}$ and $G_{3}$ that provides the difference in the connectivity measures? Obviously, we can describe it in terms of internally disjoint paths, but there must be some more basic property that separates them. In particular, notice how vertex $c$ seems to be the connecting point between the two halves of $G_{3}$; that it, $c$ is a cut-vertex.
Theorem 4.17 A graph $G$ with at least 3 vertices is 2-connected if and only if $G$ is connected and does not have any cut-vertices.

## 数学代写|图论代写graph theory代考|2-Edge-Connected

Based on previous discussions, it shouldn’t be surprising that there are similar notions for graphs that are 2-edge-connected, which can be described as those graphs that are connected but without a bridge. In particular, we extend the ear decomposition idea into its edge analog, called a closed-ear decomposition.
Definition 4.25 A closed-ear in a graph $G$ is a cycle where all vertices have degree 2 in $G$ except for one vertex on the cycle. A closed-ear decomposition is a collection $P_{0}, P_{1}, \ldots P_{k}$ so that $P_{0}$ is a cycle, $P_{i}$ is either an ear or closed-ear of $P_{0} \cup \cdots \cup P_{i-1}$ for all $i \geq 1$, and all edges and vertices are included in the collection.

The small change in our definition between an ear and closed-ear can most easily be attributed to graphs that are 2-edge-connected, but not 2-connected. Consider the graph from Example 4.4. This graph is 2-edge-connected since it does not have a bridge. Thus the same decomposition we had above still works (since we are allowed to use ears in a closed-ear decomposition). In contrast, the graph $G_{7}$ below (sometimes called the bow-tie graph) is 2-edge-connected but not 2-connected since $c$ is a cut-vertex. If we tried to find a regular ear decomposition for $G_{7}$ then we would run into a problem in finding $P_{1}$ since once the first cycle has been chosen (for example $P_{0}$ below) then the remaining portion of the graph would only consist of another 3 -cycle. But allowing $P_{1}$ to be a closed-ear, we find our closed-ear decomposition.

## Matlab代写

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