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# 数学代写|图论代考GRAPH THEORY代写|Cut-Vertices

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## 数学代写|图论代写GRAPH THEORY代考|Cut-Vertices

It is probably clear by now that one of the most important properties that a graph can possess is that of being connected. Figure 5.1 shows seven graphs of order 7 . The graph $G_1$ is a tree, $G_4=C_7$ and $G_7$ $=K_7$. Obviously, all of these graphs are connected. However, some appear to be “more connected” than others. Indeed, the main goal of this chapter is the introduction of measures of how connected a graph is.

Some graphs are so slightly connected that the removal of a single edge results in a disconnected graph. We have already seen this and an edge with this property is a bridge. The graph $G_2$ has a bridge. So does $G_1$. In fact, every edge of $G_1$ is a bridge since $G_1$ is a tree. We now turn from connected graphs containing an edge whose removal results in a disconnected graph to connected graphs containing a vertex whose removal results in a disconnected graph.

Recall that if $v$ is a vertex of a nontrivial graph $G$, then by $G-v$ we mean the (induced) subgraph of $G$ whose vertex set consists of all vertices of $G$ except $v$ and whose edge set consists of all edges of $G$ except those incident with $v$. This concept is illustrated in Figure 5.2. In fact, if $U$ is a proper subset of the vertex set of $G$, then $G-U$ is the (induced) subgraph of $G$ whose vertex set is $V(G)-U$ and whose edge set consists of all edges of $G$ joining two vertices in $V(G)-U$. A vertex $v$ in a connected graph $G$ is a cut-vertex of $G$ if $G-v$ is disconnected. More generally, a vertex $v$ is a cutvertex in a graph $G$ if $v$ is a cut-vertex of a component of $G$. In the graph $G$ of Figure 5.2, $v$ and $x$ are the only cut-vertices. In the graph $G-v$, the vertex $x$ is not a cut-vertex; however, $s$ is a cut-vertex of $G-v$. Consequently, for $U={s, v}$, the graph $G-U$ is disconnected. The graphs $G_1, G_2$ and $G_3$ of Figure 5.1 also contain cut-vertices but no other graphs in Figure 5.1 contain cut-vertices.

## 数学代写|图论代写GRAPH THEORY代考|Blocks

We now turn our attention from connected graphs containing cut-vertices to connected graphs that contain no cut-vertices. A nontrivial connected graph with no cut-vertices is called a nonseparable graph. Hence all of the graphs $G_4, G_5, G_6$ and $G_7$ of Figure 5.1 are nonseparable. In addition, $K_2$ is a nonseparable graph; indeed, $K_2$ is the only nonseparable graph of order 2. Since nonseparable graphs of order 3 or more contain no cut-vertices, they contain no bridges; that is, every edge lies on a cycle. In fact, more can be said.

Theorem 5.7 A graph of order at least 3 is nonseparable if and only if every two vertices lie on a common cycle.

Proof. First, suppose that $G$ is a graph of order at least 3 such that every two vertices of $G$ lie on a common cycle. Assume, to the contrary, that $G$ is not nonseparable. Since every two vertices lie on a common cycle, every two vertices are connected and so $G$ is connected. Because $G$ is not nonseparable, $G$ must contain a cut-vertex, say $v$. Let $u$ and $w$ be two vertices that belong to different components of $G-v$. By assumption, $u$ and $w$ lie on a common cycle $C$ on $G$. However then, $C$ determines two distinct $u-w$ paths of $G$, at least one of which does not contain $v$, contradicting Theorem 5.3. Therefore, $G$ contains no cut-vertices and so $G$ is nonseparable.

## MATLAB代写

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