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# 数学代写|图论代考GRAPH THEORY代写|2-Edge-Connected

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## 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.

The edge analog to Theorem 4.24 is given below and has a very similar proof (see Exercise 4.31).

Theorem 4.26 A graph $G$ is 2-edge-connected if and only if it has a closed-ear decomposition.

The majority of this chapter has been devoted to a very theoretical aspect of graph theory. The remainder of this chapter looks at various applications of connectivity, with the largest focus on network flow.

## 数学代写|图论代写GRAPH THEORY代考|Network Flow

Digraphs have appeared throughout this text to model asymmetric relationships. For example, at the beginning of Chapter 1 , we described game wins as a directed edge in a tournament, and in Chapter 2 we looked at when digraphs, more specifically tournaments, were hamiltonian. This section will focus on a new application for digraphs, one in which items are sent through a network. These networks often model physical systems, such as sending water through pipelines or information through a computer network. The digraphs we investigate will need a starting and ending location, though there is no requirement for the network to be acyclic. In this section, we will need some specialized terminology, in particular, what we mean by a network.

Definition 4.27 A network is a digraph where each arc $e$ has an associated nonnegative integer $c(e)$, called a capacity. In addition, the network has a designated starting vertex $\boldsymbol{s}$, called the source, and a designated ending vertex $\boldsymbol{t}$, called the $\operatorname{sink}$. A flow $f$ is a function that assigns a value $f(e)$ to each arc of the network.

Below is an example of a network. Each arc is given a two-part label. The first component is the flow along the arc and the second component is the capacity.

The names of the starting and ending vertices are reminiscent of a system of pipes with water coming from the source, traveling through some configuration of the piping to arrive at the sink (ending vertex). Using this analogy further, we can see that some restraints need to be placed on the flow along an arc. For example, flow should travel in the indicated direction of the arcs, no arc can carry more than its capacity, and the amount entering a junction point (a vertex) should equal the amount leaving. These rules are more formally stated in the following definitions.

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