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第16行：
第16行： − + 第29行：
第29行： − + − + − + − − + − '''<font color="#ff8000">步道 Walk</font>'''是连接一系列顶点形成的有限或无限的边序列 第44行：
第42行： − + − * A '''trail''' is a walk in which all edges are distinct.{{sfn|Bender|Williamson|2010|p=162}} − '''<font color="#ff8000">轨迹 Trail</font>'''是所有的边缘都清晰可见的步道 − + − + + 第59行：
第56行： − − − Some authors do not require that all vertices of a path be distinct and instead use the term '''simple path''' to refer to such a path. − − Some authors do not require that all vertices of a path be distinct and instead use the term simple path to refer to such a path. + − A [[weighted graph]] associates a value (''weight'') with every edge in the graph. The ''weight of a walk'' (or trail or path) in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words ''cost'' or ''length'' are used instead of weight. − − A weighted graph associates a value (weight) with every edge in the graph. The weight of a walk (or trail or path) in a weighted graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight. − − 一个'''<font color="#ff8000">加权图 Weighted Graph</font>'''将一个值(权)与图中的每条边相关联。一个加权图中的步道(或轨迹或路径)的权是所有边的权之和。有时，“成本”或“长度”这两个词可以用来指代“权重”这一概念。 − + − + − '''<font color="#ff8000">有向步道 Directed Walk</font>'''是指由连接一系列顶点的边沿相同方向定向形成的有限或无限序列。 − − Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a directed graph. A finite directed walk is a sequence of edges for which there is a sequence of vertices such that for . is the vertex sequence of the directed walk. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or ray) has a first vertex but no last vertex. − + − '''<font color="#ff8000">有向轨迹 Directed Trail</font>'''是指所有边都可见的轨迹。+ − + − '''<font color="#ff8000">有向路径 Directed Path</font>'''是指所有顶点都不同的有向路径。+ − − If is a finite directed walk with vertex sequence then w is said to be a walk from v<sub>1</sub> to v<sub>n</sub>. Similarly for a directed trail or a path. If there is a finite directed walk between two distinct vertices then there is also a finite directed trail and a finite directed path between them. − + − Some authors do not require that all vertices of a directed path be distinct and instead use the term '''simple directed path''' to refer to such a directed path. − − Some authors do not require that all vertices of a directed path be distinct and instead use the term simple directed path to refer to such a directed path. − − 有些学者并不要求有向路径的所有顶点都是不同的，因此他们用“简单有向路径”这个术语来表示它。 第113行：
第91行： − 一个加权有向图将一个值(权)与有向图中的每条边相关联。加权有向图中有向步道(或路径)的权是有向边权重的和。有时，“成本”或“长度”这两个词可以用来指代“权重”这一概念。+ − + − + − − + − − + − − 如果两条路径没有任何共同的内部顶点，那么它们与顶点无关(换言之，内部顶点不相交)。类似地，如果两条路径没有任何共同的内边，那么它们是边独立的(或边不相交)。两条内部顶点不相交的路径的边也不相交，但反之未必正确的。+ + + − + − 图中两个顶点之间，如果最短路径存在，那么它们的的距离是最短路径的长度，否则该距离是无穷大。 − * The diameter of a connected graph is the largest distance (defined above) between pairs of vertices of the graph. − 连接图的直径是图的顶点对之间的最大距离（如上定义）。 − + 第144行：
第118行： − 求解图的最短路径和最长路径的均有多种算法，但是前者的计算比后者简单得多。+
无编辑摘要
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.
In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction.
在'''<font color="#ff8000">图论 Graph Theory</font>'''中，图中的'''<font color="#ff8000">路径 Path</font>'''是一个有限或无限的边序列，这些边连接着一系列顶点，这些顶点在大多数定义中都是不同的(又因为这些顶点是不同的，所以图中的边也是不同的)。'''<font color="#ff8000">有向图 Directed Graph</font>'''(有时也称为dipath)中的有向路径，在具有附加条件：所有的边都方向相同时是一个有限或无限的边序列，它连接一系列不同的顶点。
在[[图论 Graph theory]]中，图中的'''路径 path'''是一个有限或无限的边序列，这些边连接着一系列顶点，这些顶点在大多数定义中都是不同的(又因为这些顶点是不同的，所以图中的边也是不同的)。有向图(有时也称为dipath)中的有向路径，在具有附加条件：所有的边都方向相同时是一个有限或无限的边序列，它连接一系列不同的顶点。
--[[用户:CecileLi|CecileLi]]([[用户讨论:CecileLi|讨论]]) 【审校】“邦迪和默蒂(1976) ，吉本斯(1985) ，或迪斯特尔(2005)。柯尔特(1990)等人著作中，”一句改为“如在邦迪和默蒂(1976) 、吉本斯(1985) 、迪斯特尔(2005)、柯尔特(1990)等人著作中，”
--[[用户:CecileLi|CecileLi]]([[用户讨论:CecileLi|讨论]]) 【审校】“邦迪和默蒂(1976) ，吉本斯(1985) ，或迪斯特尔(2005)。柯尔特(1990)等人著作中，”一句改为“如在邦迪和默蒂(1976) 、吉本斯(1985) 、迪斯特尔(2005)、柯尔特(1990)等人著作中，”
== Definitions 定义==
== 定义==
=== Walk, trail, path 步道 轨迹 路径===
===步道 轨迹 路径 Walk, trail, path ===
[[File:Trail but not path.svg|200px|thumb|right
[[File:Trail but not path.svg|200px|thumb|right |图2：Trail from A to E，but not path 从A到E的轨迹而非路径]]
图2：Trail from A to E，but not path 从A到E的轨迹而非路径]]
* A '''walk''' is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] which joins a sequence of [[Vertex (graph theory)|vertices]].{{sfn|Bender|Williamson|2010|p=162}}
* '''步道 Walk'''：是连接一系列[[顶点]]形成的有限或无限的边序列。{{sfn|Bender|Williamson|2010|p=162}}
Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a graph. A finite walk is a sequence of edges {{nowrap|(''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}} for which there is a sequence of vertices {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} such that {{nowrap begin}}''ϕ''(''e''<sub>''i''</sub>) = {''v''<sub>''i''</sub>, ''v''<sub>''i'' + 1</sub>}{{nowrap end}} for {{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} is the ''vertex sequence'' of the walk. This walk is ''closed'' if {{nowrap begin}}''v''<sub>1</sub> = ''v''<sub>''n''</sub>{{nowrap end}}, and ''open'' else. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or [[End (graph theory)|ray]]) has a first vertex but no last vertex.
Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a graph. A finite walk is a sequence of edges {{nowrap|(''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}} for which there is a sequence of vertices {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} such that {{nowrap begin}}''ϕ''(''e''<sub>''i''</sub>) = {''v''<sub>''i''</sub>, ''v''<sub>''i'' + 1</sub>}{{nowrap end}} for {{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} is the ''vertex sequence'' of the walk. This walk is ''closed'' if {{nowrap begin}}''v''<sub>1</sub> = ''v''<sub>''n''</sub>{{nowrap end}}, and ''open'' else. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or [[End (graph theory)|ray]]) has a first vertex but no last vertex.
Let be a graph. A finite walk is a sequence of edges for which there is a sequence of vertices such that ϕ(e<sub>i</sub>) = {v<sub>i</sub>, v<sub>i + 1</sub>} for . is the vertex sequence of the walk. This walk is closed if v<sub>1</sub> = v<sub>n</sub>, and open else. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or ray) has a first vertex but no last vertex.
Let be a graph. A finite walk is a sequence of edges for which there is a sequence of vertices such that ϕ(e<sub>i</sub>) = {v<sub>i</sub>, v<sub>i + 1</sub>} for . is the vertex sequence of the walk. This walk is closed if v<sub>1</sub> = v<sub>n</sub>, and open else. An infinite walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite walk (or ray) has a first vertex but no last vertex.
以一个图为例{{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} 。有限步道是一系列的边{{nowrap|(''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}}，其顶点序列{{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}}。 {{nowrap begin}}''ϕ''(''e''<sub>''i''</sub>) = {''v''<sub>''i''</sub>, ''v''<sub>''i'' + 1</sub>}{{nowrap end}}对于{{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} 是移动的顶点序列。如果 {{nowrap begin}}''v''<sub>1</sub> = ''v''<sub>''n''</sub>{{nowrap end}} ，则此步道封闭，反之则开放。一个无限步道是由一系列边组成的，它们的类型与这里描述的相同，但没有起点或终点，而一个半无限步道(或光线)则有起点但是没有终点。
以一个图为例{{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} 。'''有限步道 finite walk'''是一系列的边{{nowrap|(''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}}，其顶点序列{{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}}。 {{nowrap begin}}''ϕ''(''e''<sub>''i''</sub>) = {''v''<sub>''i''</sub>, ''v''<sub>''i'' + 1</sub>}{{nowrap end}}对于{{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} 是移动的顶点序列。如果 {{nowrap begin}}''v''<sub>1</sub> = ''v''<sub>''n''</sub>{{nowrap end}} ，则此步道封闭，反之则开放。'''无限步道 infinite walk'''是由一系列边组成的，它们的类型与这里描述的相同，但没有起点或终点，而一个半无限步道(或光线)则有起点但是没有终点。
* A '''path''' is a trail in which all vertices (and therefore also all edges) are distinct.{{sfn|Bender|Williamson|2010|p=162}}
* '''轨迹 trail'''是所有的边缘都清晰可见的步道。{{sfn|Bender|Williamson|2010|p=162}}
'''<font color="#ff8000">路径 Path</font>'''是一条所有顶点(因此也包括所有边)都是不同的轨迹。
* '''路径 path'''是一条所有顶点(因此也包括所有边)都是不同的轨迹。{{sfn|Bender|Williamson|2010|p=162}}
如果 {{nowrap|1=''w'' = (''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}}是有顶点的序列 {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}}，那么 w 就是从 ''v''<sub>1</sub> ''到'' ''v''<sub>''n''</sub>的有限步道。同样，对于一条轨迹或者一条路径也是如此。如果两个不同的顶点之间有一个有限步道，那么在它们之间也有一条有限的轨迹和一条有限的路径。
如果 {{nowrap|1=''w'' = (''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}}是有顶点的序列 {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}}，那么 w 就是从 ''v''<sub>1</sub> ''到'' ''v''<sub>''n''</sub>的有限步道。同样，对于一条轨迹或者一条路径也是如此。如果两个不同的顶点之间有一个有限步道，那么在它们之间也有一条有限的轨迹和一条有限的路径。
有些学者对路径的定义中并不要求它的所有顶点都是不同的，而是使用术语“简单路径”来指代这样的路径。
有些学者对路径的定义中并不要求它的所有顶点都是不同的，而是使用术语“简单路径”来指代这样的路径。
一个'''加权图 weighted graph'''将一个值（权重）与图中的每条边相关联。一个加权图中的步道（或轨迹或路径）的权是所有边的权之和。有时，“成本 cost”或“长度 length”这两个词可以用来指代“权重”这一概念。
=== Directed walk, trail, path 有向的步道，轨迹，路径===
=== 有向的步道，轨迹，路径 Directed walk, trail, path===
* A '''directed walk''' is a finite or infinite [[sequence]] of [[Edge (graph theory)|edges]] directed in the same direction which joins a sequence of [[Vertex (graph theory)|vertices]].{{sfn|Bender|Williamson|2010|p=162}}
* '''有向步道 directed walk'''是指由连接一系列顶点的边沿相同方向定向形成的有限或无限序列。{{sfn|Bender|Williamson|2010|p=162}}
Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a directed graph. A finite directed walk is a sequence of edges {{nowrap|(''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}} for which there is a sequence of vertices {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} such that {{nowrap|1=''ϕ''(''e''<sub>''i''</sub>) = (''v''<sub>''i''</sub>, ''v''<sub>''i'' + 1</sub>)}} for {{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} is the ''vertex sequence'' of the directed walk. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or [[End (graph theory)|ray]]) has a first vertex but no last vertex.
Let {{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}} be a directed graph. A finite directed walk is a sequence of edges {{nowrap|(''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}} for which there is a sequence of vertices {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} such that {{nowrap|1=''ϕ''(''e''<sub>''i''</sub>) = (''v''<sub>''i''</sub>, ''v''<sub>''i'' + 1</sub>)}} for {{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} is the ''vertex sequence'' of the directed walk. An infinite directed walk is a sequence of edges of the same type described here, but with no first or last vertex, and a semi-infinite directed walk (or [[End (graph theory)|ray]]) has a first vertex but no last vertex.
以一个有向图{{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}}为例。有限有向步道由一系列的边组成 {{nowrap|(''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}}，而对于这些边，又存在一系列的顶点 {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}}。{{nowrap|1=''ϕ''(''e''<sub>''i''</sub>) = (''v''<sub>''i''</sub>, ''v''<sub>''i'' + 1</sub>)}}对于{{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}}是有向步道的顶点序列。无限有向步道是一个边序列，其类型与本文描述的相同，但起点或终点，而半无限有向步道(或射线)有起点，但没有终点。
以一个有向图{{nowrap|1=''G'' = (''V'', ''E'', ''ϕ'')}}为例。有限有向步道由一系列的边组成 {{nowrap|(''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}}，而对于这些边，又存在一系列的顶点 {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}}。{{nowrap|1=''ϕ''(''e''<sub>''i''</sub>) = (''v''<sub>''i''</sub>, ''v''<sub>''i'' + 1</sub>)}}对于{{nowrap|1=''i'' = 1, 2, …, ''n'' − 1}}. {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}}是有向步道的顶点序列。无限有向步道是一个边序列，其类型与本文描述的相同，但起点或终点，而半无限有向步道(或射线)有起点，但没有终点。
* A '''directed trail''' is a directed walk in which all edges are distinct.{{sfn|Bender|Williamson|2010|p=162}}
* A '''directed trail''' is a directed walk in which all edges are distinct.
'''有向轨迹 directed trail'''是指所有边都可见的轨迹。{{sfn|Bender|Williamson|2010|p=162}}
* A '''directed path''' is a directed trail in which all vertices are distinct.{{sfn|Bender|Williamson|2010|p=162}}
* A '''directed path''' is a directed trail in which all vertices are distinct.
'''有向路径 directed path'''是指所有顶点都不同的有向路径。{{sfn|Bender|Williamson|2010|p=162}}
If {{nowrap|1=''w'' = (''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}} is a finite directed walk with vertex sequence {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} then ''w'' is said to be a ''walk from'' ''v''<sub>1</sub> ''to'' ''v''<sub>''n''</sub>. Similarly for a directed trail or a path. If there is a finite directed walk between two ''distinct'' vertices then there is also a finite directed trail and a finite directed path between them.
If {{nowrap|1=''w'' = (''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}} is a finite directed walk with vertex sequence {{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}} then ''w'' is said to be a ''walk from'' ''v''<sub>1</sub> ''to'' ''v''<sub>''n''</sub>. Similarly for a directed trail or a path. If there is a finite directed walk between two ''distinct'' vertices then there is also a finite directed trail and a finite directed path between them.
假设一个步道{{nowrap|1=''w'' = (''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}}是顶点序列有限的{{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}}有向步道，那么 w 就是从 ''v''<sub>1</sub> ''到'' ''v''<sub>''n''</sub>的步道。同样，对于有向轨迹或路径也是如此。如果在两个不同的顶点之间存在有限有向步道，那么在它们之间也存在有限有向轨迹和有限有向路径。
假设一个步道{{nowrap|1=''w'' = (''e''<sub>1</sub>, ''e''<sub>2</sub>, …, ''e''<sub>''n'' − 1</sub>)}}是顶点序列有限的{{nowrap|(''v''<sub>1</sub>, ''v''<sub>2</sub>, …, ''v''<sub>''n''</sub>)}}有向步道，那么 w 就是从 ''v''<sub>1</sub> ''到'' ''v''<sub>''n''</sub>的步道。同样，对于有向轨迹或路径也是如此。如果在两个不同的顶点之间存在有限有向步道，那么在它们之间也存在有限有向轨迹和有限有向路径。
有些学者并不要求有向路径的所有顶点都是不同的，因此他们用“简单有向路径 simple directed path”这个术语来表示它。
A weighted directed graph associates a value (weight) with every edge in the directed graph. The weight of a directed walk (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.
A weighted directed graph associates a value (weight) with every edge in the directed graph. The weight of a directed walk (or trail or path) in a weighted directed graph is the sum of the weights of the traversed edges. Sometimes the words cost or length are used instead of weight.
一个加权有向图将一个值（权重）与有向图中的每条边相关联。加权有向图中有向步道（或路径）的权是有向边权重的和。有时，“成本”或“长度”这两个词可以用来指代“权重”这一概念。
== Examples 例子==
==例子==
* A graph is [[Connectivity (graph theory)|connected]] if there are paths containing each pair of vertices.
* 如果路径包含每一对顶点，那么图就是连通的。
如果路径包含每一对顶点，那么图就是连通的。
* A directed graph is [[Strongly-connected digraph|strongly connected]] if there are oppositely oriented directed paths containing each pair of vertices.
* A directed graph is [[Strongly-connected digraph|strongly connected]] if there are oppositely oriented directed paths containing each pair of vertices.
如果一个有向图中存在包含每对顶点的相对有向路径，那么这个有向图就是紧密连通的。
如果一个有向图中存在包含每对顶点的相对有向路径，那么这个有向图就是紧密连通的。
* A path such that no graph edges connect two nonconsecutive path vertices is called an [[induced path]].
* 没有边连接两个不连续的路径顶点的路径称为'''<font color="#ff8000">诱导路径 induced path</font>'''。
没有边连接两个不连续的路径顶点的路径称为'''<font color="#ff8000">诱导路径 Induced Path</font>'''。
* A path that includes every vertex of the graph is known as a [[Hamiltonian path]].
* 包含图的每个顶点的路径称为'''<font color="#ff8000">哈密顿路径 Hamiltonian Path</font>'''。
包含图的每个顶点的路径称为'''<font color="#ff8000">哈密顿路径 Hamiltonian Path</font>'''。
* Two paths are ''vertex-independent'' (alternatively, ''internally vertex-disjoint'') if they do not have any internal vertex in common. Similarly, two paths are ''edge-independent'' (or ''edge-disjoint'') if they do not have any internal edge in common. Two internally vertex-disjoint paths are edge-disjoint, but the converse is not necessarily true.
* Two paths are ''vertex-independent'' (alternatively, ''internally vertex-disjoint'') if they do not have any internal vertex in common. Similarly, two paths are ''edge-independent'' (or ''edge-disjoint'') if they do not have any internal edge in common. Two internally vertex-disjoint paths are edge-disjoint, but the converse is not necessarily true.
如果两条路径没有任何共同的内部顶点，那么它们与顶点无关（换言之，内部顶点不相交）。类似地，如果两条路径没有任何共同的内边，那么它们是边独立的（或边不相交）。两条内部顶点不相交的路径的边也不相交，但反之未必正确的。
* 图中两个顶点之间，如果最短路径存在，那么它们的的距离是最短路径的长度，否则该距离是无穷大。
* The [[Distance (graph theory)|distance]] between two vertices in a graph is the length of a shortest path between them, if one exists, and otherwise the distance is infinity.
* 连接图的直径是图的顶点对之间的最大距离（如上定义）。
== Finding paths 路径寻找==
== 路径寻找 Finding paths==
Several algorithms exist to find [[Shortest path problem|shortest]] and [[Longest path problem|longest]] paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.
Several algorithms exist to find [[Shortest path problem|shortest]] and [[Longest path problem|longest]] paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.
Several algorithms exist to find shortest and longest paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.
Several algorithms exist to find shortest and longest paths in graphs, with the important distinction that the former problem is computationally much easier than the latter.
求解图的[[最短路径]]和[[最长路径]]的均有多种算法，但是前者的计算比后者简单得多。