introduction

Following the previous article, after understanding the basic knowledge related to graph theory, let's take a look at a few classic problems in graph theory.

2. Minimum spanning tree problem

2.1 Definition of tree and its basic properties

A connected graph without cycles is called a tree.

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It has several fundamental properties that are interrelated.

The nature of the tree 1 - if $T$ is a tree, and the number of points in the tree $n_T \geq 2$ , thenThere are at least two suspension points in $T.$

The nature of the tree 2 - Figure $T$ is a tree, thenThe number of sides in $T$ $m$ is equal to the number of pointsSubtract 1 from $n$ $m = n - 1.$

The nature of the tree 3 - Figure $The necessary and sufficient condition for T$ to be a tree is that there is exactly one link between any two vertices.

From property 3, the following inferences can be drawn:

If any edge is removed from a tree, the remaining graph is disconnected. It can be seen that in all graphs with the same set of points, the tree is the connected graph with the fewest edge trees.
Fill in an edge between two non-adjacent points in the tree, and you get exactly one circle.

In summary, it can be concluded that the tree Six basic properties of $T , namely:$

acyclic graph;
connected graph;
$m_T=n_T-1;$
minus one edge is not connected;
Adding a side has exactly one circle;
At least two hanging points.

2.2 The braced tree of a graph

Suppose graph $T = (V, E^{'})$ is graph $G = (V, E)$ the supporting subgraph, if the graph $T$ is a tree, call itA spanning tree of $G.$

Theorem - Graph The necessary and sufficient condition for $G$ $G$ is connected.

Proof: The necessity clearly holds.

sufficiency. Design $G$ is a connected graph, if it does not contain a cycle, it is a tree itself, so $G$ is a brace tree of itself. If $G$ contains a cycle, and any edge can be removed from the cycle to getA supporting subgraph $G_1$ $of G$ $G_{1}$ . if $G_1$ Does not contain circles, then $G_1$ is $A spanning tree of G$ ; if $G_1$ contains a circle, then from $G_1$ Take any circle in the circle, remove another edge from the circle, and get $G_2$ , and so on, and finally a support subgraph $G_k without circles can be obtained$ , the brace tree.

The proof of the sufficiency of this theorem also inspires us a way to find spanning trees of connected graphs. It is to take a circle at random, remove one side from the circle, and repeat this step for the remaining circles until no circle is contained, and a support tree can be obtained. This method is called "the method of breaking circles " .

You can also think in reverse, take any edge $e_1 from the graph$ , find a line that does not match $e_1$ $e_2$ that make up the circle $e_{2}$ , find another one that does not match $e_1,e_2)$ constitutes the side $e_3$ , until it becomes impossible. At this time, the graph formed by all the edges taken out is a braced tree, and this method is called " circle avoidance method ".

2.3 Definition of minimum spanning tree and related theorems

Given an undirected network $G = (V, E, W)$ , for $\in E$ in $G$ $e \in E$ , design $\geq0$ , forEvery spanning tree $of G$ $T$ , we defineThe value of this $W ( T$ $W(T)=\sum_{e\in E}w(e)$ if the spanning tree $T^*$ is the one with the smallest weight among all supporting trees, then it is called $T^*$ isThe minimum spanning tree of $G.$

Theorem - Let $T$ is the network $G = (V, E, W)$ support tree, any edge $\in E$ , uniquely determines a circle $C (e)$ . ExclusionOutside $e$ All other edges of $C$ $($ $e$ $)$ $T$ , if $T$ is forA minimum spanning tree of $G$ $e$ isMaximum weight edges in $C$ $($ $e$ $) .$

Proof: In the cycle Take any other side $in C$ $($ $e$ $)$ $e^{'}$ ，刪 $\cup\{e\}-\{e'\}$ is still connected and does not contain circles, that is, it is a tree, denoted as $T^{'}$ ,then $W (T^{'}) = W (T) + w (e) - w (ie^{'})$ . Thanks to $T$ is the minimum spanning tree, $\leq W(T')$ ，即 $\leq w(e')$ , since $e^{’}$ , so the original proposition is proved.

Theorem - Let $T$ is the network $G = (V, E, W)$ support tree, then for any edge $\in E$ , uniquely determinesA cut set of $G$ $\varPhi(e)$ , $e$ 外， $\varPhi(e)$ None of the other edges on $Φ$ $($ $e$ $)$ $T$ , if $T$ is forThe minimum spanning tree of $G$ $e$ is $\varPhi(e)$ Minimum weight edges in $Φ$ $($ $e$ $) .$

2.4 Minimum spanning tree algorithm

There are three main algorithms, which are the circle avoidance method, the anti circle method and the circle breaking method.

2.4.1 Circle avoidance method (KRUSKAL algorithm)

First set the network $The edges in G$ are sorted according to their weight, starting from the smallest edge, each time a new edge is selected, it is necessary to judge whether it forms a circle with the selected edge, if so, discard the edge, otherwise the selected edge knows all The number of selected edges is equal to the number of vertices minus 1. Each time an edge is selected, the edge with the smallest weight is given priority, so it is also called the minimum edge optimization method.

2.4.2 Anti-circle method (PRIM algorithm)

From graph $G = (V, E)$ any vertex is taken and put into the treeThe point set $V_T$ $of T$ $V_{T}$ , for the graph $G$ is divided into $V_T,\overline{V}_T$ The cut set $\varPhi(V_T)$ , select an edge with the smallest weight and put it into the tree $Set the edges of T$ , and put the vertices associated with the edges into $V_T$ , repeat the above steps until all vertices are selected into $V_T$ until.

2.4.3 Circle breaking method

Let $G^{(k)}$ isThe connected generation subgraph of $G$ $GG^{(0)}=G$ ), if $G^{(k)}$ does not contain a circle, then it is a minimum spanning tree; if it contains a circle, select the edge with the largest weight in the circle to remove, and repeat the above process until no circle is found.

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Next question, let's take a look at the shortest path problem!

[Management Operations Research] Chapter 7 | Graph and Network Analysis (2, Minimum Spanning Tree Problem)

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