# Pendant Vertices, Non-Pendant Vertices, Pendant Edges and Non-Pendant Edges in Graph

Pre-requisites: Handshaking theorem.

__Pendant Vertices__

__Pendant Vertices__

Let **G** be a graph, A vertex **v** of **G** is called a **pendant vertex** if and only if **v** has **degree 1**. In other words, pendant vertices are the vertices that have **degree 1**, also called **pendant vertex**.

Note:Degree = number of edges connected to a vertex

In the case of trees, a **pendant vertex** is known as a **terminal node** or **leaf node**, or **leaf** since it has only **1** degree. Remember leaf nodes have only 1 degree so pendant vertex is called a leaf node in the case of trees.

**Example:** In the given diagram **A** and **B** are pendant vertices because each of them has degree **1**.

Let’s take this example to print all the pendant vertices in the graph.

## C++

`// C++ program for the above approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to print all the pendant` `// vertices` `void` `printPendantVertices(map<` `char` `, vector<` `char` `> > graph)` `{` ` ` `// All the vectors which contain only 1` ` ` `// vertex i.e, size 1 has only 1 edge` ` ` `// hence a pendant vertex.` ` ` `for` `(` `auto` `x : graph) {` ` ` `if` `(x.second.size() == 1) {` ` ` `cout << x.first << ` `" "` `;` ` ` `}` ` ` `}` `}` `// Driver Code` `int` `main()` `{` ` ` `map<` `char` `, vector<` `char` `> > graph;` ` ` `graph[` `'A'` `].push_back(` `'B'` `);` ` ` `graph[` `'B'` `].push_back(` `'A'` `);` ` ` `graph[` `'C'` `].push_back(` `'B'` `);` ` ` `graph[` `'B'` `].push_back(` `'C'` `);` ` ` `printPendantVertices(graph);` ` ` `return` `0;` `}` |

## Python3

`# Python program for the above approach` `# Function to print all the pendant` `# vertices` `def` `printPendantVertices(graph):` ` ` `# All the vectors which contain only 1` `# vertex i.e, size 1 has only 1 edge` `# hence a pendant vertex.` ` ` `for` `x,y ` `in` `graph.items():` ` ` `if` `(` `len` `(y) ` `=` `=` `1` `):` ` ` `print` `(x, end ` `=` `" "` `)` `# Driver Code` `graph ` `=` `{}` `graph[` `"A"` `] ` `=` `[` `"B"` `]` `graph[` `"B"` `] ` `=` `[` `"A"` `]` `graph[` `"C"` `] ` `=` `[` `"B"` `]` `graph[` `"B"` `].append(` `"A"` `)` `printPendantVertices(graph)` `# This code is contributed by shinjanpatra` |

## Javascript

`<script>` `// Javascript program for the above approach` `// Function to print all the pendant` `// vertices` `function` `printPendantVertices(graph) {` ` ` `// All the vectors which contain only 1` ` ` `// vertex i.e, size 1 has only 1 edge` ` ` `// hence a pendant vertex.` ` ` `for` `(x of graph) {` ` ` `if` `(x[1].length == 1) {` ` ` `document.write(x[0] + ` `" "` `);` ` ` `}` ` ` `}` `}` `// Driver Code` `let graph = ` `new` `Map();` `graph.set(` `"A"` `, [` `"B"` `]);` `graph.set(` `"B"` `, [` `"A"` `]);` `graph.set(` `"C"` `, [` `"B"` `]);` `graph.set(` `"B"` `, [...graph.get(` `"B"` `)].push(` `"A"` `));` `printPendantVertices(graph);` `// This code is contributed by saurabh_jaiswal.` `</script>` |

**Output**

A C

__Non-Pendant Vertices__

__Non-Pendant Vertices__

Non-Pendant Vertices are the vertices that have **degrees **other than** 1**. In the case of trees, a non-pendant vertex is a **non-leaf node**, since it does not have degree **1** ( leaf nodes have degree **1**).

**Example: **In the given diagram **A** and **C** are pendant vertices because **A** and **C** have degree **1**, B is a non-pendant vertex because it contains degrees other than **1** which is **2**.

__Pendant Edge__

__Pendant Edge__

An edge of a graph is said to be a pendant edge if and only if **one of its vertices is a pendant vertex.**

**Example:** In the given diagram **AB** is a pendant edge since it contains pendant vertex as one of its vertexes.

__Non-Pendant Edge__

__Non-Pendant Edge__

An edge of a graph is said to be a non-pendant edge if **it does not contain a** **pendant vertex as one of its vertexes.**

**Example:** in the given diagram **AB** is a pendant edge since it has pendant vertex (A) as one of its vertexes. **BD, BC, DC** are non-pendant vertices.

__Examples:__

__Examples:__

Let us see some questions based examples on the above topics:

**Q1. Suppose that a tree T has 2 vertices of degree 2, 4 vertices of degree 3**, **and 3 vertices of degree 4. find the number of pendant vertices in T.**

Finding number pendant vertices is nothing but finding the number of leaf nodes.

Let’s use the Handshaking Theorem formulaSum of all degrees = 2 * (Sum of Edges)

(2 vertices) * (2 degrees) + (4 vertices) * (3 degrees) + (3 vertices) * (4 degrees) + (k vertices) * (1 degree) = (2 * edges)

where

kis pendant vertices or leaf nodes which have degree 1

eis total number of edges in the tree2*2 + 4*3 + 3*4 + k*1 = 2*e ——-(1)

remember :

number of edges = number of vertices – 1e=(2+4+3+k)-1

e=9+k-1

e=8+k ——-(2)

putting equation 2 in 1 gives

4 + 12 + 12 + k = 2(8+k)

28 + k = 16 + 2k

-2k + k = 16 – 28

-k = -12

k = 12

so total number of pendant vertices are 12

**Q2. If a tree T has 4 vertices of degree 2, 1 vertex of degree 3 and 2 vertices of degree 4 and 1 vertex of degree 5. find the number of pendant vertices in T.**

Finding number pendant vertices is nothing but finding the number of leaf nodes.

Let’s use the Handshaking Theorem formula

Sum of all degrees = 2 * (Sum of Edges)

(4 vertices)*(2 degrees) + (1 vertex)*(3 degrees) + (2 vertices)*(4 degrees) + (1 vertex)*(5 degree) + (k vertices)*(1 degree) = (2 * edges)

where

kis pendant vertices or leaf nodes which have degree 1

eis total number of edges in the tree4*2 + 1*3 + 2*4 + 1*5 + k*1 = 2*e ——-(1)

remember number of edges = number of vertices – 1

e=(4+1+2+1+k)-1

e=8+k-1

e=7+k ——-(2)

putting equation 2 in 1 gives

8 + 3 + 8 + 5 + k = 2(7+k)

24 + k = 14 + 2k

-2k + k = 14 – 24

-k = -10

k = 10

so total number of pendant vertices are 10