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+
+# 17 ͼ
+
+**֪ʶ½á¹¹£º**
+
+
+![ͼ1 ֪ʶ½á¹¹](https://img-blog.csdnimg.cn/20201108112251166.png)
+
+
+---
+
+## 1. ͼµÄ»ù±¾¸ÅÄîÓëÊõÓï
+
+### 1.1 ͼµÄ¶¨Òå
+
+ͼÓɶ¥µã¼¯ºÍ±ß¼¯×é³É£¬¼ÇΪ $G=(V,E)$¡£
+- ¶¥µã¼¯£º¶¥µãµÄÓÐÇî·Ç¿Õ¼¯ºÏ£¬¼ÇΪ$V(G)$¡£
+- ±ß¼¯£º¶¥µãż¶ÔµÄÓÐÇºÏ£¬¼ÇΪ$E(G)$ ¡£
+
+±ß£º
+- ÎÞÏò±ß£º$(v_i,v_j)=(v_j,v_i)$ 
+- ÓÐÏò±ß£¨»¡£©£º$ <v_i,v_j> \neq <v_j,v_i> $ ʼµã$v_i$³ÆΪ»¡Î²£¬ÖÕµã$v_j$³ÆΪ»¡Í·¡£
+
+ÀýÌ⣺
+
+![ͼ2 ͼµÄ±íʾ](https://img-blog.csdnimg.cn/20201106202532801.png)
+ 
+
+ÀýÌ⣺
+
+![ͼ3 ͼµÄ±íʾ](https://img-blog.csdnimg.cn/20201106202637934.png)
+
+### 1.2 ͼµÄ·ÖÀà
+
+- ÓÐÏòͼ£º$E(G)$ÓÉÓÐÏò±ß¹¹³É¡£
+- ÎÞÏòͼ£º$E(G)$ÓÉÎÞÏò±ß¹¹³É¡£
+
+### 1.3 ͼÖж¥µãÊýÓë±ßÊýµÄ¹Øϵ
+
+£¨1£©É趥µãÊýΪ$n$£¬±ßÊýΪ$e$Ôò£º
+
+- ÎÞÏòͼ£º$0\leq e \leq \frac{n(n-1)}{2}$£¬$e=\frac{n(n-1)}{2}$µÄͼ³ÆΪÎÞÏòÍêȫͼ¡£
+- ÓÐÏòͼ£º$0\leq e \leq n(n-1)$£¬$e = n(n-1)$µÄͼ³ÆΪÓÐÏòÍêȫͼ¡£
+
+£¨2£©¶¥µãµÄ¶È
+
+- ÎÞÏòͼ£º$D(v)=$Óë$v$Á¬½ÓµÄ±ßÊý¡£ 
+- ÓÐÏòͼ£º$D(v)=ID(v)+OD(v)$ 
+    - $ID(v)$£ºÒÔ$v$ΪÖյ㣨»¡Í·£©µÄ±ßÊý¡£
+    - $OD(v)$£ºÒÔ$v$ΪÆðµã£¨»¡Î²£©µÄ±ßÊý¡£
+
+£¨3£©¶ÈÓë±ßµÄ¹Øϵ 
+
+
+$$
+e=\frac{\sum_{i=1}^{n}D(v_i)}{2}
+$$
+
+£¨4£©ÕýÔòͼ£ºËùÓж¥µãµÄ¶È¶¼ÏàͬµÄͼ
+ 
+![ͼ4 ÕýÔòͼ](https://img-blog.csdnimg.cn/20201106203517590.png)
+
+
+### 1.4 ·¾¶
+
+£¨1£©¶¨Ò壺ͼ$G=(V,E)$£¬Èô´æÔÚÒ»¸ö¶¥µãÐòÁÐ $v_p,v_{i1},v_{i2},\cdots,v_{im},v_q$£¬Ê¹µÃ$(v_p,v_{i1}),(v_{i1},v_{i2}),\cdots,(v_{im},v_q)\in E(G)$£¨ÎÞÏòͼ£©»òÕß$ <v_p,v_{i1}>,<v_{i1},v_{i2}>,\cdots,<v_{im},v_q>\in E(G)$£¨ÓÐÏòͼ£©Ôò³Æ$v_p$µ½$v_q$´æÔÚÒ»Ìõ·¾¶¡£
+
+£¨2£©Â·¾¶µÄ³¤¶È£ºÂ·¾¶ÉϱߵÄÊýÄ¿¡£
+
+£¨3£©¼òµ¥Â·¾¶£ºÈôÒ»Ìõ·¾¶Éϳý$v_p$£¬$v_q$¿ÉÏàͬÍ⣬ÆäÓඥµã¾ù²»ÏàͬµÄ·¾¶¡£
+
+£¨4£©¼òµ¥»Ø·£º$v_p$Óë$v_q$ÏàͬµÄ¼òµ¥Â·¾¶¡£
+
+### 1.5 ×Óͼ
+Éè$G=(V,E)$ÊÇÒ»¸öͼ£¬Èô$V'\subseteq V$£¬$E'\subseteq E$ÇÒ$E^{'}$ÖеıßËù¹ØÁªµÄ¶¥µã$\in V'$£¬Ôò$G'=(V',E')$ Ò²ÊÇÒ»¸öͼ£¬²¢³ÆÆäΪ$G$µÄ×Óͼ¡£
+ 
+![ͼ5 ͼÓë×Óͼ](https://img-blog.csdnimg.cn/20201106204617379.png)
+
+### 1.6 Á¬Í¨Í¼ÓëÁ¬Í¨·ÖÁ¿£¨ÎÞÏòͼ£©
+
+Á¬Í¨Í¼£ºÎÞÏòͼ$G$µÄ$V(G)$ÖУ¬ÈκÎÁ½¸ö¶¥µã¶¼´æÔÚ·¾¶£¬Ôò³Æ$G$ΪÁ¬Í¨Í¼¡£
+
+Á¬Í¨·ÖÁ¿£ºÎÞÏòͼ$G$µÄ¼«´óÁ¬Í¨×Óͼ¡£
+
+×¢£º
+
+- Á¬Í¨Í¼µÄÁ¬Í¨·ÖÁ¿Ö»ÓÐÒ»¸ö£¬ÎªÆä±¾Éí¡£
+- ·ÇÁ¬Í¨Í¼µÄÁ¬Í¨·ÖÁ¿²»Î¨Ò»¡£
+
+Àý×Ó£º
+ 
+![ͼ6 Á¬Í¨Í¼ÓëÁ¬Í¨·ÖÁ¿](https://img-blog.csdnimg.cn/20201106204848603.png)
+
+### 1.7 Ç¿Á¬Í¨Í¼ÓëÇ¿Á¬Í¨·ÖÁ¿£¨ÓÐÏòͼ£©
+
+Ç¿Á¬Í¨Í¼£ºÓÐÏòͼ$G$µÄ$V(G)$ÖУ¬ÈκÎÁ½¸ö¶¥µã¶¼´æÔÚ·¾¶£¬Ôò³Æ$G$ΪǿÁ¬Í¨Í¼¡£
+
+Ç¿Á¬Í¨·ÖÁ¿£ºÓÐÏòͼ$G$µÄ¼«´óÁ¬Í¨×Óͼ¡£
+
+×¢£º
+- Ç¿Á¬Í¨Í¼µÄÇ¿Á¬Í¨·ÖÁ¿Ö»ÓÐÒ»¸ö£¬ÎªÆä±¾Éí¡£
+- ·ÇÇ¿Á¬Í¨Í¼µÄÇ¿Á¬Í¨·ÖÁ¿²»Î¨Ò»¡£
+
+Àý×Ó£º
+ 
+![ͼ7 Ç¿Á¬Í¨Í¼ÓëÇ¿ÁªÍ¨·ÖÁ¿](https://img-blog.csdnimg.cn/20201106205035361.png)
+
+### 1.8 ÍøÂç
+
+$E(G)$ÖеÄÿÌõ±ß¶¼´øÓÐȨֵµÄͼ³ÆΪÍøÂç¡£
+
+×¢£ºÈ¨Ò»°ã¾ßÓÐʵ¼ÊÒâÒ壬±ÈÈç¾àÀë¡¢ÏûºÄµÈ¡£
+
+Àý×Ó£º
+ 
+![ͼ8 ÍøÂç](https://img-blog.csdnimg.cn/20201106205129363.png)
+
+
+---
+## 2. ͼµÄ´æ´¢½á¹¹
+
+### 2.1 ˳Ðò´æ´¢£¨ÁÚ½Ó¾ØÕó£©
+
+ÁÚ½Ó¾ØÕ󣺱íʾ¶¥µãÖ®¼äÏàÁÚ¹ØϵµÄ¾ØÕó
+
+Èôͼ$G=(V,E)$£¬ÔòÆäÁÚ½Ó¾ØÕó¿É±íʾΪ£º
+
+$$
+A[i,j]=\begin{cases}
+1, & (v_i,v_j)\in E(G) or \left\langle v_i,v_j \right\rangle  \in E(G)\\
+0, & (v_i,v_j)\notin E(G) or \left\langle v_i,v_j \right\rangle \notin E(G)
+\end{cases}
+$$
+
+Àý×Ó£ºÎÞÏòͼµÄ´æ´¢
+ 
+![ͼ9 ÁÚ½Ó¾ØÕó´æ´¢ÎÞÏòͼ](https://img-blog.csdnimg.cn/20201106205902391.png)
+
+
+Àý×Ó£ºÓÐÏòͼµÄ´æ´¢
+ 
+![ͼ10 ÁÚ½Ó¾ØÕó´æ´¢ÓÐÏòͼ](https://img-blog.csdnimg.cn/20201106205934703.png)
+
+Èôͼ$G=(V,E)$ΪÍøÂ磬ÔòÆäÁÚ½Ó¾ØÕó¿É±íʾΪ£º
+
+
+$$
+A[i,j]=\begin{cases}
+\omega_{ij}, & (v_i,v_j)\in E(G) or \left\langle v_i,v_j \right\rangle\in E(G)\\
+0 or \infty, & (v_i,v_j)\notin E(G) or \left\langle v_i,v_j \right\rangle\notin E(G)
+\end{cases}
+$$
+
+Àý×Ó£ºÍøÂçµÄ´æ´¢
+ 
+![ͼ11 ÁÚ½Ó¾ØÕó´æ´¢ÍøÂç](https://img-blog.csdnimg.cn/20201106210322224.png)
+ 
+![ͼ12 MGraph½á¹¹](https://img-blog.csdnimg.cn/20201106210702707.png)
+
+ 
+```c
+using System;
+namespace NonLinearStruct.Graph
+{
+    /// <summary>
+    /// ͼ³éÏóÊý¾ÝÀàÐÍʵÏÖ--ÀûÓÃÁÚ½Ó¾ØÕó
+    /// </summary>
+    public class MGraph
+    {
+        private readonly double[,] _adMatrix;//ÁÚ½Ó¾ØÕó
+        private readonly string[] _vertexNameList;//´æ´¢Í¼Öи÷½áµãÃû×ÖµÄÊý×é
+
+        /// <summary>
+        /// »ñȡͼÖнáµãµÄ¸öÊý
+        /// </summary>
+        public int VertexCount { get; }
+
+        /// <summary>
+        /// ³õʼ»¯MGraphÀàµÄÐÂʵÀý
+        /// </summary>
+        /// <param name="vCount">ͼÖнáµãµÄ¸öÊý</param>
+        public MGraph(int vCount)
+        {
+            if (vCount <= 0)
+                throw new ArgumentOutOfRangeException();
+
+            VertexCount = vCount;
+            _vertexNameList = new string[vCount];
+            _adMatrix = new double[vCount, vCount];
+        }
+        
+        /// <summary>
+        /// »ñÈ¡»òÉèÖÃͼÖи÷½áµãµÄÃû³Æ
+        /// </summary>
+        /// <param name="index">½áµãÃû³Æ´ÓÁ㿪ʼµÄË÷Òý</param>
+        /// <returns>Ö¸¶¨Ë÷Òý´¦½áµãµÄÃû³Æ</returns>
+        public string this[int index]
+        {
+            get
+            {
+                if (index < 0 || index > VertexCount - 1)
+                    throw new IndexOutOfRangeException();
+                return _vertexNameList[index];
+            }
+            set
+            {
+                if (index < 0 || index > VertexCount - 1)
+                    throw new IndexOutOfRangeException();
+                _vertexNameList[index] = value;
+            }
+        }
+        
+        private int GetIndex(string vertexName)
+        {
+            //¸ù¾Ý½áµãÃû×Ö»ñÈ¡½áµãÔÚÊý×éÖеÄϱê
+            int i;
+            for (i = 0; i < VertexCount; i++)
+            {
+                if (_vertexNameList[i] == vertexName)
+                    break;
+            }
+            return i == VertexCount ? -1 : i;
+        }
+        
+        /// <summary>
+        /// ¸øÁÚ½Ó¾ØÕó¸³Öµ
+        /// </summary>
+        /// <param name="startVertexName">Æðʼ½áµãµÄÃû×Ö</param>
+        /// <param name="endVertexName">ÖÕÖ¹½áµãµÄÃû×Ö</param>
+        /// <param name="weight">±ßÉϵÄȨֵ»òÁ¬½Ó¹Øϵ</param>
+        public void AddEdge(string startVertexName, string endVertexName, double weight = 1)
+        {
+            int i = GetIndex(startVertexName);
+            int j = GetIndex(endVertexName);
+            if (i == -1 || j == -1)
+                throw new Exception("ͼÖв»´æÔڸñß.");
+            _adMatrix[i, j] = weight;
+        }
+    }
+}
+```
+
+
+
+### 2.2 Á´Ê½´æ´¢£¨ÁÚ½Ó±í£©
+
+ÁÚ½Ó±í£ºÓÉ˳Ðò´æ´¢µÄ¶¥µã±íºÍÁ´Ê½´æ´¢µÄ±ß±í¹¹³ÉµÄͼµÄ´æ´¢½á¹¹¡£
+
+Àý×Ó£ºÎÞÏòͼµÄ´æ´¢
+ 
+![ͼ13 ÎÞÏòͼµÄ´æ´¢](https://img-blog.csdnimg.cn/20201106210912939.png)
+
+Àý×Ó£ºÓÐÏòͼµÄ´æ´¢
+
+![ͼ14 ÓÐÏòͼµÄ´æ´¢](https://img-blog.csdnimg.cn/2020110621102015.png)
+
+
+Àý×Ó£ºÍøÂçµÄ´æ´¢
+
+![ͼ15 ÍøÂçµÄ´æ´¢](https://img-blog.csdnimg.cn/2020110621111817.png)
+ 
+±ß±í½áµãµÄʵÏÖ£º
+ 
+![ͼ16 ±ß±í½áµã½á¹¹](https://img-blog.csdnimg.cn/20201106211252949.png)
+
+![ͼ17 EdgeNode½á¹¹](https://img-blog.csdnimg.cn/20201106211415100.png)
+
+```c
+using System;
+namespace NonLinearStruct.Graph
+{
+    /// <summary>
+    /// ÁÚ½Ó±í±ß±íÉϵĽáµã
+    /// </summary>
+    public class EdgeNode
+    {
+        /// <summary>
+        /// »ñÈ¡±ßÖÕµãÔÚ¶¥µãÊý×éÖеÄλÖÃ
+        /// </summary>
+        public int Index { get; }
+
+        /// <summary>
+        /// »ñÈ¡±ßÉϵÄȨֵ
+        /// </summary>
+        public double Weight { get; }
+
+        /// <summary>
+        /// »ñÈ¡»òÉèÖÃÏÂÒ»¸öÁÚ½Óµã
+        /// </summary>
+        public EdgeNode Next { get; set; }
+
+        /// <summary>
+        /// ³õʼ»¯EdgeNodeÀàµÄÐÂʵÀý
+        /// </summary>
+        /// <param name="index">±ßÖÕµãÔÚ¶¥µãÊý×éÖеÄλÖÃ</param>
+        /// <param name="weight">±ßÉϵÄȨֵ</param>
+        /// <param name="next">ÏÂÒ»¸öÁÚ½Óµã</param>
+        public EdgeNode(int index, double weight = 0.0, EdgeNode next = null)
+        {
+            if (index < 0)
+                throw new ArgumentOutOfRangeException();
+
+            Index = index;
+            Weight = weight;
+            Next = next;
+        }
+    }
+}
+```
+
+¶¥µã±í½áµãµÄʵÏÖ£º
+ 
+![ͼ18 ¶¥µã±í½áµã½á¹¹](https://img-blog.csdnimg.cn/20201106211512903.png)
+
+![ͼ19 VertextNode½á¹¹](https://img-blog.csdnimg.cn/20201106211657496.png)
+ 
+```c
+namespace NonLinearStruct.Graph
+{
+    /// <summary>
+    /// ÁÚ½Ó±í¶¥µã±íÖеĽáµã
+    /// </summary>
+    public class VertexNode
+    {
+        /// <summary>
+        /// »ñÈ¡»òÉèÖö¥µãµÄÃû×Ö
+        /// </summary>
+        public string VertexName { get; set; }
+
+        /// <summary>
+        /// »ñÈ¡»òÉèÖö¥µãÊÇ·ñ±»·ÃÎÊ
+        /// </summary>
+        public bool Visited { get; set; }
+
+        /// <summary>
+        /// »ñÈ¡»òÉèÖö¥µãµÄµÚÒ»¸öÁÚ½Óµã
+        /// </summary>
+        public EdgeNode FirstNode { get; set; }
+
+        /// <summary>
+        /// ³õʼ»¯VertexNodeÀàµÄÐÂʵÀý
+        /// </summary>
+        /// <param name="vName">¶¥µãµÄÃû×Ö</param>
+        /// <param name="firstNode">¶¥µãµÄµÚÒ»¸öÁÚ½Óµã</param>
+        public VertexNode(string vName, EdgeNode firstNode = null)
+        {
+            VertexName = vName;
+            Visited = false;
+            FirstNode = firstNode;
+        }
+    }
+}
+```
+
+ͼ½á¹¹µÄʵÏÖ
+
+![ͼ20 AdGraph½á¹¹](https://img-blog.csdnimg.cn/20201106211947741.png)
+ 
+```c
+using System;
+using System.Collections.Generic;
+using System.Linq;
+using LinearStruct;
+
+namespace NonLinearStruct.Graph
+{
+    /// <summary>
+    /// ͼ³éÏóÊý¾ÝÀàÐÍʵÏÖ--ÀûÓÃÁÚ½Ó±í
+    /// </summary>
+    public class AdGraph
+    {
+        private readonly VertexNode[] _vertexList; //½áµã±í
+
+        /// <summary>
+        /// »ñȡͼµÄ½áµãÊý
+        /// </summary>
+        public int VertexCount { get; }
+
+        /// <summary>
+        /// ³õʼ»¯AdGraphÀàµÄÐÂʵÀý
+        /// </summary>
+        /// <param name="vCount">ͼÖнáµãµÄ¸öÊý</param>
+        public AdGraph(int vCount)
+        {
+            if (vCount <= 0)
+                throw new ArgumentOutOfRangeException();
+
+            VertexCount = vCount;
+            _vertexList = new VertexNode[vCount];
+        }
+
+        /// <summary>
+        /// »ñÈ¡»òÉèÖÃͼÖи÷½áµãµÄÃû³Æ
+        /// </summary>
+        /// <param name="index">½áµãÃû³Æ´ÓÁ㿪ʼµÄË÷Òý</param>
+        /// <returns>Ö¸¶¨Ë÷Òý´¦½áµãµÄÃû³Æ</returns>
+        public string this[int index]
+        {
+            get
+            {
+                if (index < 0 || index > VertexCount - 1)
+                    throw new ArgumentOutOfRangeException();
+
+                return _vertexList[index] == null
+                    ? "NULL"
+                    : _vertexList[index].VertexName;
+            }
+            set
+            {
+                if (index < 0 || index > VertexCount - 1)
+                    throw new ArgumentOutOfRangeException();
+
+                if (_vertexList[index] == null)
+                    _vertexList[index] = new VertexNode(value);
+                else
+                    _vertexList[index].VertexName = value;
+            }
+        }
+
+        /// <summary>
+        /// µÃµ½½áµãÔÚ½áµã±íÖеÄλÖÃ
+        /// </summary>
+        /// <param name="vertexName">½áµãµÄÃû³Æ</param>
+        /// <returns>½áµãµÄλÖÃ</returns>
+        private int GetIndex(string vertexName)
+        {
+            int i;
+            for (i = 0; i < VertexCount; i++)
+            {
+                if (_vertexList[i] != null && _vertexList[i].VertexName == vertexName)
+                    break;
+            }
+            return i == VertexCount ? -1 : i;
+        }
+
+        /// <summary>
+        /// ¸øͼ¼Ó±ß
+        /// </summary>
+        /// <param name="startVertexName">Æðʼ½áµãµÄÃû×Ö</param>
+        /// <param name="endVertexName">ÖÕÖ¹½áµãµÄÃû×Ö</param>
+        /// <param name="weight">±ßÉϵÄȨֵ</param>
+        public void AddEdge(string startVertexName, string endVertexName, double weight = 0.0)
+        {
+            int i = GetIndex(startVertexName);
+            int j = GetIndex(endVertexName);
+
+            if (i == -1 || j == -1)
+                throw new Exception("ͼÖв»´æÔڸñß.");
+
+            EdgeNode temp = _vertexList[i].FirstNode;
+            if (temp == null)
+            {
+                _vertexList[i].FirstNode = new EdgeNode(j, weight);
+            }
+            else
+            {
+                while (temp.Next != null)
+                    temp = temp.Next;
+                temp.Next = new EdgeNode(j, weight);
+            }
+        }
+    }
+}
+```
+
+---
+## 3. ͼµÄ±éÀú
+### 3.1 Éî¶ÈÓÅÏÈËÑË÷
+
+Ëã·¨£º
+
+¼ÙÉè$G=(V,E)$ÒÔ$S$ΪÆðµã£¨Ô´µã£©½øÐÐËÑË÷¡£
+
+Ê×Ïȱêʶ$S$ΪÒÑ·ÃÎʽáµã£¬½Ó×ÅÑ°ÕÒÓë$S$ÏàÁڵĽáµã$\omega$£¬Èô$\omega$ÊÇδ±»·ÃÎʽáµã£¬ÔòÒÔ$\omega$ΪÆðµã½øÐÐÉî¶ÈÓÅÏÈËÑË÷£¬Èô$\omega$ÊÇÒѱ»·ÃÎʽáµã£¬ÔòÑ°ÕÒÆäËüÓë$S$ÏàÁڵĽáµã£¬Ö±µ½Óë$S$Óз¾¶ÏàͨµÄËùÓнáµã¶¼±»·ÃÎʹý¡£
+
+Àý×Ó£º
+
+![ͼ21 ͼµÄ½á¹¹](https://img-blog.csdnimg.cn/20201106212721439.png)
+ 
+Éî¶ÈÓÅÏÈËÑË÷µÄÐòÁÐΪ£º$V_0,V_1,V_3,V_7,V_4,V_5,V_2,V_6$ 
+
+ËäÈ»±éÀúÐòÁв»Î¨Ò»£¬µ«ÊÇÁÚ½Ó±íÈ·¶¨ºó£¬±éÀúÐòÁоÍΨһÁË¡£
+
+ʵÏÖ£º
+
+![ͼ22 AdGraph½á¹¹](https://img-blog.csdnimg.cn/20201106213043485.png)
+
+ 
+
+```c
+private void Dfs(int i, ref string dfsResult)
+{
+    //Éî¶ÈÓÅÏÈËÑË÷µÝ¹éº¯Êý
+    _vertexList[i].Visited = true;
+    dfsResult += _vertexList[i].VertexName + "\n";
+
+    EdgeNode p = _vertexList[i].FirstNode;
+    while (p != null)
+    {
+        if (_vertexList[p.Index].Visited)
+            p = p.Next;
+        else
+            Dfs(p.Index, ref dfsResult);
+    }
+}
+
+/// <summary>
+/// µÃµ½Éî¶ÈÓÅÏÈËÑË÷ÐòÁÐ
+/// </summary>
+/// <param name="startVertexName">½øÐÐÉî¶ÈÓÅÏÈËÑË÷µÄÆðʼµãÃû³Æ</param>
+/// <returns>Éî¶ÈÓÅÏÈËÑË÷ÐòÁÐ</returns>
+public string DfsTraversal(string startVertexName)
+{
+    string dfsResult = string.Empty;
+    int i = GetIndex(startVertexName);
+    if (i != -1)
+    {
+        for (int j = 0; j < VertexCount; j++)
+            _vertexList[j].Visited = false;
+        Dfs(i, ref dfsResult);
+    }
+    return dfsResult;
+}
+```
+
+### 3.2 ¹ã¶ÈÓÅÏÈËÑË÷
+
+Ëã·¨£º
+
+¼ÙÉè$G=(V,E)$ÒÔ$S$ΪÆðµã£¨Ô´µã£©½øÐÐËÑË÷¡£
+
+Ê×Ïȱêʶ$S$ΪÒÑ·ÃÎʽáµã£¬½Ó×Å·ÃÎÊ$S$µÄÁÚ½Óµã$\omega_1,\omega_2,\cdots,\omega_t$È»ºó·ÃÎÊ$\omega_1,\omega_2,\cdots,\omega_t$µÄδ±»·ÃÎʵÄÁڽӵ㣬ÒÔ´ËÀàÍÆ£¬Ö±µ½Óë$S$Óз¾¶ÏàÁ¬µÄËùÓнáµã¶¼±»·ÃÎʵ½¡£
+
+ÏÈÀ´ÏÈ·þÎñµÄ˼Ïë¡£
+
+Àý×Ó£º
+
+![ͼ23 ͼµÄ½á¹¹](https://img-blog.csdnimg.cn/20201106213215211.png)
+
+Éî¶ÈÓÅÏÈËÑË÷µÄÐòÁÐΪ£º$V_0,V_1,V_2,V_3,V_4,V_5,V_6,V_7$
+
+ËäÈ»±éÀúÐòÁв»Î¨Ò»£¬µ«ÊÇÁÚ½Ó±íÈ·¶¨ºó£¬±éÀúÐòÁоÍΨһÁË¡£
+
+
+ʵÏÖ£º
+
+![ͼ24 AdGraph½á¹¹](https://img-blog.csdnimg.cn/20201106213551207.png)
+
+```c
+/// <summary>
+/// µÃµ½¹ã¶ÈÓÅÏÈËÑË÷ÐòÁÐ
+/// </summary>
+/// <param name="startNodeName">½øÐйã¶ÈÓÅÏÈËÑË÷µÄÆðʼµãÃû³Æ</param>
+/// <returns>¹ã¶ÈÓÅÏÈËÑË÷ÐòÁÐ</returns>
+public string BfsTraversal(string startNodeName)
+{
+    string bfsResult = string.Empty;
+    int i = GetIndex(startNodeName);
+    if (i != -1)
+    {
+        for (int j = 0; j < VertexCount; j++)
+            _vertexList[j].Visited = false;
+
+        _vertexList[i].Visited = true;
+        bfsResult += _vertexList[i].VertexName + "\n";
+        LinkQueue<int> lq = new LinkQueue<int>();
+        lq.EnQueue(i);
+        while (lq.IsEmpty() == false)
+        {
+            int j = lq.QueueFront;
+            lq.DeQueue();
+            EdgeNode p = _vertexList[j].FirstNode;
+            while (p != null)
+            {
+                if (_vertexList[p.Index].Visited == false)
+                {
+                    _vertexList[p.Index].Visited = true;
+                    bfsResult += _vertexList[p.Index].VertexName + "\n";
+                    lq.EnQueue(p.Index);
+                }
+                p = p.Next;
+            }
+        }
+    }
+    return bfsResult;
+}
+```
+
+---
+## 4. ÍØÆËÅÅÐò
+
+### 4.1 »ù±¾¸ÅÄî
+
+$AOV$Íø£¨Activity on Vertex Network£©£ºÓö¥µã±íʾ»î¶¯£¬ÓÃÓÐÏò±ß±íʾ»î¶¯Ö®¼äÏȺó¹ØϵµÄÓÐÏòͼ¡£
+
+ÍØÆËÐòÁУº°Ñ$AOV$ÍøÖеÄËùÓж¥µãÅųÉÒ»¸öÏßÐÔÐòÁУ¬Ê¹Ã¿¸ö»î¶¯µÄËùÓÐÇ°Çý»î¶¯¶¼ÅÅÔڸûµÄÇ°±ß¡£
+
+ÍØÆËÅÅÐò£º¹¹Ôì$AOV$ÍøÍØÆËÐòÁеĹý³Ì¡£
+
+ÀýÌ⣺°´ÕÕÍØÆËÅÅÐò°²ÅÅÏÂÁпγÌ
+
+![ͼ25 ¿Î³Ì±í](https://img-blog.csdnimg.cn/20201107142129131.png)
+
+![ͼ26 AOVÍø](https://img-blog.csdnimg.cn/2020110714223963.png)
+
+
+![ͼ27 AOVÍøµÄ´æ´¢](https://img-blog.csdnimg.cn/20201107142312104.png)
+ 
+ 
+### 4.2 Ëã·¨²½Öè
+
+µÚ1²½£º´ÓÍøÖÐÑ¡ÔñÒ»¸öÈë¶ÈΪ0µÄ¶¥µã¼ÓÈëÅÅÐòÐòÁС£
+
+µÚ2²½£º´ÓÍøÖÐɾ³ý¸Ã¶¥µã¼°ÆäËùÓгö±ß¡£
+
+µÚ3²½£ºÖ´ÐеÚ1¡¢2²½£¬Ö±µ½ËùÓж¥µãÒÑÅÅÐò»òÍøÖÐÊ£ÓඥµãÈë¶È¾ù²»Îª0£¨ËµÃ÷£ºÍøÖдæÔڻط£¬ÎÞ·¨¼ÌÐøÍØÆËÅÅÐò£©¡£
+
+ÍØÆËÐòÁУº
+
+£¨1£©$V_0,V_1,V_4,V_6,V_2,V_3,V_7,V_5,V_8$
+
+£¨2£©$V_0,V_6,V_1,V_2,V_3,V_4,V_5,V_8,V_7$
+
+×¢£º¶ÔÓÚÈκÎÎ޻طµÄ$AOV$Íø£¬Æ䶥µã¾ù¿ÉÅųÉÍØÆËÐòÁУ¬µ«ÆäÍØÆËÐòÁÐδ±ØΨһ¡£
+
+![ͼ28 ´æÔڻطµÄAOVÍø](https://img-blog.csdnimg.cn/20201107142605197.png)
+
+´æÔڻطµÄÓÐÏòͼ£¬²»Äܹ¹³ÉÍØÆËÐòÁС£
+
+### 4.3 Ë㷨ʵÏÖ
+
+![ͼ29 AdGraph½á¹¹](https://img-blog.csdnimg.cn/20201107144220299.png)
+
+```c
+/// <summary>
+/// µÃµ½Ã¿¸ö½ÚµãµÄÈë¶È
+/// </summary>
+/// <returns></returns>
+private int[] GetInDegressList()
+{
+    int[] id = new int[VertexCount];
+    for (int i = 0; i < VertexCount; i++)
+    {
+        EdgeNode p = _vertexList[i].FirstNode;
+        while (p != null)
+        {
+            id[p.Index]++;
+            p = p.Next;
+        }
+    }
+    return id;
+}
+
+/// <summary>
+/// µÃµ½AOVÍøµÄÍØÆËÅÅÐòÐòÁÐ
+/// </summary>
+/// <returns>AOVÍøµÄÍØÆËÅÅÐòÐòÁÐ</returns>
+public string TopoSort()
+{
+    string result = string.Empty;
+    int[] id = GetInDegressList();
+    LinkQueue<int> lq = new LinkQueue<int>();
+    for (int i = 0; i < VertexCount; i++)
+    {
+        if (id[i] == 0)
+            lq.EnQueue(i);
+    }
+
+    if (lq.Length == VertexCount)
+        throw new Exception("´ËÓÐÏòͼÎÞÓÐÏò±ß.");
+
+    while (lq.IsEmpty() == false)
+    {
+        int j = lq.QueueFront;
+        lq.DeQueue();
+        result += _vertexList[j].VertexName + "\n";
+
+        EdgeNode p = _vertexList[j].FirstNode;
+        while (p != null)
+        {
+            id[p.Index]--;
+            if (id[p.Index] == 0)
+            {
+                lq.EnQueue(p.Index);
+            }
+            p = p.Next;
+        }
+    }
+    int k;
+    for (k = 0; k < VertexCount; k++)
+    {
+        if (id[k] != 0)
+        {
+            break;
+        }
+    }
+    return (k == VertexCount) ? result : "¸ÃAOVÍøÓл·.";
+}
+```
+
+
+---
+## 5. ×îСÉú³ÉÊ÷
+
+### 5.1 »ù±¾¸ÅÄî
+
+Éú³ÉÊ÷£ºÉè$G$ΪÁ¬Í¨Íø£¬¾ßÓÐ$G$µÄËùÓж¥µã£¨$n$¸ö£©ÇÒÖ»ÓÐ$n-1$Ìõ±ßµÄÁ¬Í¨×ÓÍø¡£
+
+Ê÷µÄȨ£ºÉú³ÉÊ÷$T$µÄ¸÷±ßµÄȨֵ×ܺ͡£
+
+×îСÉú³ÉÊ÷£ºÈ¨Öµ×îСµÄÉú³ÉÊ÷¡£
+
+### 5.2 PrimËã·¨
+
+Ëã·¨Ô­Àí£¨Ì°ÐÄËã·¨£©£º
+
+Éè$G=(V,E)$ΪÁ¬Í¨Íø£¬$T=(U,TE)$ΪÆä¶ÔÓ¦µÄ×îСÉú³ÉÊ÷£¬´Ó$v_0$¿ªÊ¼¹¹Ôì¡£
+
+£¨1£©¿ªÊ¼Ê±£¬$U = \lbrace u_0=v_0 \rbrace$£¬$TE=\Phi$¡£
+
+£¨2£©ÕÒµ½Âú×ã$weight(u,v)=min \lbrace weight(u_i,v_j)|u_i \in U,v_j\in V-U \rbrace$µÄ±ß£¬°ÑËü²¢Èë$TE$ÖУ¬$v$ͬʱ²¢Èë$U$¡£
+
+£¨3£©·´¸´Ö´ÐУ¨2£©£¬Ö±µ½$U=V$ʱ£¬ÖÕÖ¹Ëã·¨¡£
+
+ÀýÌ⣺
+
+![ͼ30 Á¬Í¨Í¼](https://img-blog.csdnimg.cn/20201107144310220.png)
+
+![ͼ31 Á¬Í¨Í¼µÄ´æ´¢](https://img-blog.csdnimg.cn/20201107144446641.png)
+
+![ͼ32 ×îСÉú³ÉÊ÷µÄ¹¹½¨](https://img-blog.csdnimg.cn/2020110714453692.png)
+
+
+Ë㷨ʵÏÖ£º
+
+×îСÉú³ÉÊ÷½áµãµÄ´æ´¢£º
+
+![ͼ33 ×îСÉú³ÉÊ÷½áµãµÄ½á¹¹](https://img-blog.csdnimg.cn/20201107144815248.png)
+
+
+![ͼ34 SpanTreeNode½á¹¹](https://img-blog.csdnimg.cn/20201107145414373.png)
+
+
+
+
+
+
+
+
+
+
+
+
+
+```c
+using System;
+
+namespace NonLinearStruct
+{
+    /// <summary>
+    /// Éú³ÉÊ÷½áµãµÄ³éÏóÊý¾ÝÀàÐÍ
+    /// </summary>
+    public class SpanTreeNode
+    {
+        /// <summary>
+        /// »ñÈ¡»òÉèÖýáµã±¾ÉíµÄÃû³Æ
+        /// </summary>
+        public string SelfName { get; }
+
+        /// <summary>
+        /// »ñÈ¡»òÉèÖýáµãË«Ç×µÄÃû³Æ
+        /// </summary>
+        public string ParentName { get; }
+
+        /// <summary>
+        /// »ñÈ¡»òÉèÖñߵÄȨֵ
+        /// </summary>
+        public double Weight { get; set; }
+
+        /// <summary>
+        /// ¹¹ÔìSpanTreeNodeʵÀý
+        /// </summary>
+        /// <param name="selfName">½áµã±¾ÉíµÄÃû³Æ</param>
+        /// <param name="parentName">½áµãË«Ç×µÄÃû³Æ</param>
+        /// <param name="weight">±ßµÄȨֵ</param>
+        public SpanTreeNode(string selfName, string parentName, double weight)
+        {
+            if (string.IsNullOrEmpty(selfName) || string.IsNullOrEmpty(parentName))
+                throw new ArgumentNullException();
+
+            SelfName = selfName;
+            ParentName = parentName;
+            Weight = weight;
+        }
+    }
+}
+```
+
+![ͼ35 AdGraph½á¹¹](https://img-blog.csdnimg.cn/2020110714553012.png)
+
+```c
+/// <summary>
+/// µÃµ½Á¬Í¨ÍøµÄ×îСÉú³ÉÊ÷PrimËã·¨
+/// </summary>
+/// <param name="vName">Ê÷¸ù½áµã</param>
+/// <returns>Á¬Í¨ÍøµÄ×îСÉú³ÉÊ÷</returns>
+public SpanTreeNode[] MiniSpanTree(string vName)
+{
+    int i = GetIndex(vName);
+    if (i == -1)
+        return null;
+
+    SpanTreeNode[] spanTree = new SpanTreeNode[VertexCount];
+    spanTree[0] = new SpanTreeNode(_vertexList[i].VertexName, "NULL", 0.0);
+
+    //UÖнáµãµ½¸÷½áµã×îСȨֵÄǸö½áµãÔÚVertexListÖеÄË÷ÒýºÅ
+    int[] vertexIndex = new int[VertexCount];
+    //UÖнáµãµ½¸÷¸ö½áµãµÄ×îСȨֵ
+    double[] lowCost = new double[VertexCount];
+    for (int j = 0; j < VertexCount; j++)
+    {
+        lowCost[j] = double.MaxValue;
+        vertexIndex[j] = i;
+    }
+
+    EdgeNode p1 = _vertexList[i].FirstNode;
+    while (p1 != null)
+    {
+        lowCost[p1.Index] = p1.Weight;
+        p1 = p1.Next;
+    }
+    vertexIndex[i] = -1;
+
+    for (int count = 1; count < VertexCount; count++)
+    {
+        double min = double.MaxValue;
+        int v = i;
+        for (int k = 0; k < VertexCount; k++)
+        {
+            if (vertexIndex[k] != -1 && lowCost[k] < min)
+            {
+                min = lowCost[k];
+                v = k;
+            }
+        }
+        spanTree[count] = new SpanTreeNode(_vertexList[v].VertexName, _vertexList[vertexIndex[v]].VertexName, min);
+        vertexIndex[v] = -1;
+
+        EdgeNode p2 = _vertexList[v].FirstNode;
+        while (p2 != null)
+        {
+            if (vertexIndex[p2.Index] != -1 && p2.Weight < lowCost[p2.Index])
+            {
+                lowCost[p2.Index] = p2.Weight;
+                vertexIndex[p2.Index] = v;
+            }
+            p2 = p2.Next;
+        }
+    }
+    return spanTree;
+}
+```
+
+### 5.3 KruskarËã·¨
+
+Éè$G=(V,E)$ΪÁ¬Í¨Íø£¬$T$ΪÆä¶ÔÓ¦µÄ×îСÉú³ÉÊ÷¡£
+
+£¨1£©³õʼʱ$T=\lbrace V, \lbrace \Phi \rbrace \rbrace$£¬¼´$T$ÖÐûÓбߣ¬Ö»ÓÐ$n$¸ö¶¥µã£¬¾ÍÊÇ$n$¸öÁ¬Í¨·ÖÁ¿¡£
+
+£¨2£©ÔÚ$E$ÖÐÑ¡ÔñȨֵ×îСµÄ±ß$(u,v)$£¬²¢½«´Ë±ß´Ó$E$ÖÐɾ³ý¡£
+
+£¨3£©Èç¹û$u$£¬$v$ÔÚ$T$µÄ²»Í¬µÄÁ¬Í¨·ÖÁ¿ÖУ¬Ôò½«$(u,v)$¼ÓÈëµ½$T$ÖУ¬´Ó¶øµ¼ÖÂ$T$ÖмõÉÙÒ»¸öÁ¬Í¨·ÖÁ¿¡£
+
+£¨4£©·´¸´Ö´ÐУ¨2£©£¨3£©Ö±µ½$T$ÖнöÊ£Ò»¸öÁ¬Í¨·ÖÁ¿Ê±£¬ÖÕÖ¹²Ù×÷¡£
+
+ÀýÌ⣺
+
+![ͼ36 ×îСÉú³ÉÊ÷µÄ¹¹½¨](https://img-blog.csdnimg.cn/20201107174120263.png)
+
+
+![ͼ37 Á¬Í¨Í¼µÄ´æ´¢](https://img-blog.csdnimg.cn/20201107174257367.png)
+
+![ͼ38 ±ßµÄ¼¯ºÏ](https://img-blog.csdnimg.cn/20201107174426131.png)
+
+Ë㷨ʵÏÖ£º
+
+![ͼ39 Edge½á¹¹](https://img-blog.csdnimg.cn/20201107174553303.png)
+
+
+```c
+namespace NonLinearStruct.Graph
+{
+    /// <summary>
+    /// ±íʾͼµÄ±ß
+    /// </summary>
+    public class Edge
+    {
+        /// <summary>
+        /// Æðµã±àºÅ
+        /// </summary>
+        public int Begin { get; }
+
+        /// <summary>
+        /// ÖÕµã±àºÅ
+        /// </summary>
+        public int End { get; }
+
+        /// <summary>
+        /// Ȩֵ
+        /// </summary>
+        public double Weight { get; }
+
+        /// <summary>
+        /// ´´½¨Ò»¸ö Edge ÀàµÄÐÂʵÀý
+        /// </summary>
+        /// <param name="begin">Æðµã±àºÅ</param>
+        /// <param name="end">ÖÕµã±àºÅ</param>
+        /// <param name="weight">Ȩֵ</param>
+        public Edge(int begin, int end, double weight = 0.0)
+        {
+            Begin = begin;
+            End = end;
+            Weight = weight;
+        }
+    }
+}
+```
+
+![ͼ40 AdGraph½á¹¹](https://img-blog.csdnimg.cn/2020110717501562.png)
+
+```c
+private Edge[] GetEdges()
+{
+    for (int i = 0; i < VertexCount; i++)
+        _vertexList[i].Visited = false;
+
+    List<Edge> result = new List<Edge>();
+    
+    for (int i = 0; i < VertexCount; i++)
+    {
+        _vertexList[i].Visited = true;
+        EdgeNode p = _vertexList[i].FirstNode;
+        while (p != null)
+        {
+            if (_vertexList[p.Index].Visited == false)
+            {
+                Edge edge = new Edge(i, p.Index, p.Weight);
+                result.Add(edge);
+            }
+            p = p.Next;
+        }
+    }
+    return result.OrderBy(a => a.Weight).ToArray();
+}
+
+private int Find(int[] parent, int f)
+{
+    while (parent[f] > -1)
+        f = parent[f];
+    return f;
+}
+
+/// <summary>
+/// ¿Ë³˹¿¨¶ûËã·¨ ×îСÉú³ÉÊ÷
+/// </summary>
+/// <returns></returns>
+public SpanTreeNode[] MiniSpanTree()
+{
+    int[] parent = new int[VertexCount];
+    for (int i = 0; i < VertexCount; i++)
+    {
+        parent[i] = -1;
+    }
+    SpanTreeNode[] tree = new SpanTreeNode[VertexCount];
+    Edge[] edges = GetEdges();
+
+    int count = 1;
+    for (int i = 0; i < edges.Length; i++)
+    {
+        int begin = edges[i].Begin;
+        int end = edges[i].End;
+        int b = Find(parent, begin);
+        int e = Find(parent, end);
+        if (b != e)
+        {
+            parent[e] = b;
+            tree[count] = new SpanTreeNode(_vertexList[end].VertexName,
+                _vertexList[begin].VertexName, edges[i].Weight);
+            count++;
+        }
+    }
+    for (int i = 0; i < parent.Length; i++)
+    {
+        if (parent[i] == -1)
+        {
+            tree[0] = new SpanTreeNode(_vertexList[i].VertexName, "NULL", 0.0);
+            break;
+        }
+    }
+    return tree;
+}
+```
+
+---
+## 6. µ¥Ô´×î¶Ì·¾¶
+
+### 6.1 ¶¨Òå
+Éè$G=(V,E)$ΪÁ¬Í¨Íø£¬$v$ΪԴµã£¬$v$µ½ÆäÓà¸÷¶¥µãµÄ×î¶Ì·¾¶ÎÊÌ⼴Ϊµ¥Ô´×î¶Ì·¾¶ÎÊÌâ¡£
+
+
+### 6.2 µÏ½ÜÌØ˺À­Ëã·¨
+ 
+$Dijkstra$£¨µÏ½ÜÌØ˹À­£©Ëã·¨£¨°´Â·¾¶³¤¶ÈµÝÔö˳Ðò¹¹ÔìµÄËã·¨£©£º 
+
+³õʼʱ$s$ΪԴµã£¬$D_s=0$£¬$D_i= + \infty (i \neq s)$ £¬±êʶ$s$±»·ÃÎÊ¡£
+
+£¨1£©Áî$v=s$£¬$D_v=0$£¨$v$µ½$s$µÄ·¾¶³¤¶È£©¡£
+
+£¨2£©ÒÀ´Î¿¼²ì$v$µÄδ±»·ÃÎʵÄÁÚ½Óµã$\omega$£¬Èô$D_v+weight(v,\omega)<D_\omega $£¬Ôò¸Ä±ä$D_\omega$µÄÖµ£¬Ê¹$D_\omega=D_v+weight(v,\omega )$¡£
+
+£¨3£©ÔÚδ±»·ÃÎʶ¥µãÖÐÑ¡Ôñ$D_v$×îСµÄ¶¥µã$v$£¬·ÃÎÊ$v$¡£
+
+£¨4£©Öظ´£¨2£©¡¢£¨3£©Ö±ÖÁËùÓж¥µã¶¼±»·ÃÎÊ¡£
+
+ÀýÌ⣺
+
+![ͼ41 ԭʼͼ](https://img-blog.csdnimg.cn/20201107190147852.png)
+
+![ͼ42 ͼµÄ´æ´¢½á¹¹](https://img-blog.csdnimg.cn/20201107190217139.png)
+
+![ͼ43 ×î¶Ì·¾¶½á¹û](https://img-blog.csdnimg.cn/20201107190246465.png)
+ 
+ 
+
+Ë㷨ʵÏÖ£º
+
+![ͼ44 AdGraph½á¹¹](https://img-blog.csdnimg.cn/20201107190415540.png)
+
+
+```c
+/// <summary>
+/// µ¥Ô´×î¶Ì·¾¶
+/// </summary>
+/// <param name="vName">Ñ°ÕÒ×î¶Ì·¾¶µÄÔ´µã</param>
+/// <returns>Ô´µãµ½¸÷¸ö¶¥µãµÄ×î¶Ì·¾¶</returns>
+public string ShortestPath(string vName)
+{
+    int v = GetIndex(vName);
+    if (v == -1)
+    {
+        return string.Empty;
+    }
+
+    string result = string.Empty;
+    double[] dist = new double[VertexCount];
+    string[] path = new string[VertexCount];
+    //³õʼ»¯
+    for (int i = 0; i < VertexCount; i++)
+    {
+        _vertexList[i].Visited = false;
+        dist[i] = double.MaxValue;
+        path[i] = _vertexList[v].VertexName;
+    }
+    dist[v] = 0.0;
+    _vertexList[v].Visited = true;
+
+    for (int i = 0; i < VertexCount - 1; i++)
+    {
+        EdgeNode p = _vertexList[v].FirstNode;
+        while (p != null)
+        {
+            if (_vertexList[p.Index].Visited == false
+                && dist[v] + p.Weight < dist[p.Index])
+            {
+                dist[p.Index] = dist[v] + p.Weight;
+                path[p.Index] = path[v] + " ->" + _vertexList[p.Index].VertexName;
+            }
+            p = p.Next;
+        }
+
+        double min = double.MaxValue;
+        for (int j = 0; j < VertexCount; j++)
+        {
+            if (_vertexList[j].Visited == false && dist[j] < min)
+            {
+                min = dist[j];
+                v = j;
+            }
+        }
+        _vertexList[v].Visited = true;
+    }
+
+    for (int i = 0; i < VertexCount; i++)
+    {
+        result += path[i] + ":" + dist[i] + "\n";
+    }
+
+    return result;
+}
+```
+
+
+---
+## 7. Á¬Í¨·ÖÁ¿
+
+ÀýÌ⣺
+
+![ͼ45 ԭʼͼ](https://img-blog.csdnimg.cn/2020110719092686.png)
+
+![ͼ46 ͼµÄ´æ´¢½á¹¹](https://img-blog.csdnimg.cn/20201107190953404.png)
+
+¿ÉÀûÓÃÉî¶ÈÓÅÏÈËÑË÷£¬Çó·ÇÁ¬Í¨Í¼µÄÁ¬Í¨·ÖÁ¿¡£
+
+µÚÒ»¸ö£ºA£¬B£¬D£¬C£»µÚ¶þ¸ö£ºE£¬F£»µÚÈý¸ö£ºG£¬H£¬I 
+
+Ë㷨ʵÏÖ£º
+
+![ͼ47 AdGraph½á¹¹](https://img-blog.csdnimg.cn/20201107191212105.png)
+
+```c
+/// <summary>
+/// µÃµ½Á¬Í¨·ÖÁ¿
+/// </summary>
+/// <returns>Á¬Í¨·ÖÁ¿</returns>
+public string ConnectedComponent()
+{
+    string result;
+    SLinkList<string> lst = new SLinkList<string>();
+
+    for (int i = 0; i < VertexCount; i++)
+        _vertexList[i].Visited = false;
+
+    for (int i = 0; i < VertexCount; i++)
+    {
+        if (_vertexList[i].Visited == false)
+        {
+            result = string.Empty;
+            //ÀûÓÃÉî¶ÈÓÅÏÈËÑË÷Çó·ÇÁ¬Í¨Í¼µÄÁ¬Í¨·ÖÁ¿
+            Dfs(i, ref result);
+            lst.InsertAtRear(result);
+        }
+    }
+    result = string.Empty;
+    for (int i = 0; i < lst.Length; i++)
+    {
+        result += "µÚ" + i + "¸öÁ¬Í¨·ÖÁ¿Îª:\n" + lst[i];
+    }
+    return result;
+}
+```
+
+---
+## 8. Á·Ï°
+
+¸ù¾ÝÒªÇóÍê³É³ÌÐò´úÂ룺
+
+¸ø¶¨Å¦Ô¼Êи½½üµÄÒ»·ù¼òµ¥µØͼ£¬Í¼ÖеĶ¥µãΪ³ÇÊУ¬ÎÞÏò±ß´ú±íÁ½¸ö³ÇÊеÄÁ¬Í¨¹Øϵ£¬±ßÉϵÄȨΪÁ½³ÇÊÐÖ®¼äµÄ¾àÀë¡£
+
+![ͼ48 ŦԼÊи½½üµØͼ](https://img-blog.csdnimg.cn/20201107192151266.png)
+
+£¨1£©¶Ô¸Ãͼ½øÐÐÉî¶ÈÓÅÏȺ͹ã¶ÈÓÅÏÈËÑË÷£¬²¢Êä³öËÑË÷ÐòÁУ¨Í¼µÄËÑË÷ÎÊÌ⣩¡£
+
+£¨2£©ÔÚ·ÖÎöÕâÕÅͼºó¿ÉÒÔ·¢ÏÖ£¬ÈÎÒ»¶Ô³ÇÊж¼ÊÇÁ¬Í¨µÄ¡£
+
+µÚÒ»¸öÎÊÌâÊÇ£ºÒªÓù«Â·°ÑËùÓгÇÊÐÁ¬½ÓÆðÀ´£¬ÈçºÎÉè¼Æ¿ÉʹµÃ¹¤³ÌµÄ×ÜÔì¼Û×îÉÙ£¨×îСÉú³ÉÊ÷ÎÊÌ⣩¡£
+
+µÚ¶þ¸öÎÊÌâÊÇ£ºÒª¿ª³µ´ÓÒ»¸ö³ÇÊе½ÁíÍâÒ»¸ö³ÇÊÐÇóÆä×î¶Ì¾àÀëÒÔ¼°Çý³µÂ·Ïߣ¨×î¶Ì·¾¶ÎÊÌ⣩¡£
+
+
+½â´ð£º
+
+Êý¾ÝTXTÎļþ£º
+```c
+Source,Target,Weight
+San Rapheal,Cross,12
+San Rapheal,Oakland,18
+Cross,Daly Cit,3
+Cross,San Franciso,3
+Daly Cit,San Franciso,4
+Daly Cit,Cross B,19
+San Franciso,Oakland,7
+San Franciso,San Mateo,21
+Oakland,San Iarenzo,18
+Oakland,Dublin,31
+San Iarenzo,Hayward,3
+San Iarenzo,Dublin,12
+Cross B,San Mateo,4
+Cross B,Cross C,7
+San Mateo,Hayward,13
+San Mateo,Redwood City,6
+Hayward,Freemont,9
+Dublin,San Jose,35
+Redwood City,Cross C,5
+Redwood City,Palo Alto,6
+Cross C,Cupertin,14
+Palo Alto,Freemont,9
+Palo Alto,Mtn View,6
+Freemont,San Jose,24
+Mtn View,Cupertin,6
+Mtn View,San Jose,8
+Cupertin,San Jose,7
+```
+
+
+Éî¶ÈÓÅÏÈËÑË÷ÐòÁУº
+```c
+San Rapheal
+Cross
+Daly Cit
+San Franciso
+Oakland
+San Iarenzo
+Hayward
+San Mateo
+Cross B
+Cross C
+Redwood City
+Palo Alto
+Freemont
+San Jose
+Dublin
+Mtn View
+Cupertin
+```
+
+¹ã¶ÈÓÅÏÈËÑË÷ÐòÁУº
+
+```c
+San Rapheal
+Cross
+Oakland
+Daly Cit
+San Franciso
+San Iarenzo
+Dublin
+Cross B
+San Mateo
+Hayward
+San Jose
+Cross C
+Redwood City
+Freemont
+Mtn View
+Cupertin
+Palo Alto
+```
+
+×îСÉú³ÉÊ÷£º
+
+```c
+(NULL,San Rapheal) Weight:0
+(San Rapheal,Cross) Weight:12
+(Cross,Daly Cit) Weight:3
+(Cross,San Franciso) Weight:3
+(San Franciso,Oakland) Weight:7
+(Oakland,San Iarenzo) Weight:18
+(San Iarenzo,Hayward) Weight:3
+(Hayward,Freemont) Weight:9
+(Freemont,Palo Alto) Weight:9
+(Palo Alto,Redwood City) Weight:6
+(Redwood City,Cross C) Weight:5
+(Redwood City,San Mateo) Weight:6
+(San Mateo,Cross B) Weight:4
+(Palo Alto,Mtn View) Weight:6
+(Mtn View,Cupertin) Weight:6
+(Cupertin,San Jose) Weight:7
+(San Iarenzo,Dublin) Weight:12
+
+×îСÉú³ÉÊ÷Ȩֵ£º116
+```
+
+×î¶Ì·¾¶£º
+
+```c
+San Rapheal:0
+San Rapheal ->Cross:12
+San Rapheal ->Oakland:18
+San Rapheal ->Cross ->Daly Cit:15
+San Rapheal ->Cross ->San Franciso:15
+San Rapheal ->Cross ->Daly Cit ->Cross B:34
+San Rapheal ->Cross ->San Franciso ->San Mateo:36
+San Rapheal ->Oakland ->San Iarenzo:36
+San Rapheal ->Oakland ->San Iarenzo ->Dublin:48
+San Rapheal ->Oakland ->San Iarenzo ->Hayward:39
+San Rapheal ->Cross ->Daly Cit ->Cross B ->Cross C:41
+San Rapheal ->Cross ->San Franciso ->San Mateo ->Redwood City:42
+San Rapheal ->Oakland ->San Iarenzo ->Hayward ->Freemont:48
+San Rapheal ->Cross ->San Franciso ->San Mateo ->Redwood City ->Palo Alto ->Mtn View ->San Jose:62
+San Rapheal ->Cross ->San Franciso ->San Mateo ->Redwood City ->Palo Alto:48
+San Rapheal ->Cross ->Daly Cit ->Cross B ->Cross C ->Cupertin:55
+San Rapheal ->Cross ->San Franciso ->San Mateo ->Redwood City ->Palo Alto ->Mtn View:54
+```
+