# 数学代写|图论作业代写Graph Theory代考|MS-E1050

## 数学代写|图论作业代写Graph Theory代考|Connectivity and Flow

In each of the previous chapters, we have used connectivity in the context of other problems. For example, in Chapter 2 we needed to know if a graph was connected in order to determine if it is eulerian and in Chapter 3 we define trees as minimally connected graphs, since the removal of any edge would disconnect the graph. This chapter focuses on connectivity as its own topic, where we now consider how connected a graph is, and not just whether it is connected or not. Note that this chapter is more theoretical than the previous, though we tie in network flow and end with a section on applications of connectivity.

Consider graphs $G_1, G_2$, and $G_3$ below. It should be plain to see that they are all connected graphs. We could describe other features of these graphs (are they eulerian? hamiltonian? acyclic?) but one distinguishing factor between them should be the simple difference in edge count, with $G_1$ having fewer edges than the other two. Notice that $G_2$ and $G_3$ both contain 13 edges, but their underlying structure is quite different. Visually, is seems that the edges in $G_3$ are more clumped than in $G_2$. One way to describe this clumping is in how many edges or vertices would need to be removed before the graph is no longer connected, which is one way we measure connectivity.

## 数学代写|图论作业代写Graph Theory代考|Connectivity Measures

When we define a graph to be connected, we refer to the existence of a way to move between any two vertices in a graph, specifically as the existence of a path between any pair of vertices. We will see that measuring how connected a graph is has a similar description, but we first use the standard notion described above in terms of vertex (or edge) removal.

Definition 4.1 A cut-vertex of a graph $G$ is a vertex $v$ whose removal disconnects the graph, that is, $G$ is connected but $G-v$ is not. A set $S$ of vertices within a graph $G$ is a cut-set if $G-S$ is disconnected.

Note that any connected graph that is not complete has a cut-set, whereas $K_n$ does not have a cut-set (see Exercise 4.13). Moreover, a graph can have many different cut-sets and of varying sizes. For example, two different cut-sets are shown below for graph $G_1$ above.

Although we can find many different cut-sets for graph $G_1$, we may want to choose one over another based on some sense of optimality. In particular, when we evaluate how connected a graph is, we are really asking what is the fewest number of vertices whose removal will disconnect the graph.

## 数学代写|图论作业代写Graph Theory代考|Connectivity Measures

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