Given ‘n’ vertices and ‘m’ edges of a graph. Find the minimum number and maximum number of isolated vertices that are possible in the graph.
Input : 4 2 Output : Minimum 0 Maximum 1 1--2 3--4 <---Minimum - No isolated vertex 1--2 <--- Maximum - 1 Isolated vertex i.e. 4 | 3 Input : 5 2 Output : Minimum 1 Maximum 2 1--2 3--4 5 <-- Minimum - 1 isolated vertex i.e. 5 1--2 4 5 <-- Maximum - 2 isolated vertex i.e. 4 and 5 | 3
- For minimum number of isolated vertices, we connect two vertices by only one edge. Each vertex should be only connected to one other vertex and each vertex should have degree one
Thus if the number of edges is ‘m’, and if ‘n’ vertices <=2 * 'm' edges, there is no isolated vertex and if this condition is false, there are n-2*m isolated vertices.
- For maximum number of isolated vertices, we create a polygon such that each vertex is connected to other vertex and each vertex has a diagonal with every other vertex. Thus, number of diagonals from one vertex to other vertex of n sided polygon is n*(n-3)/2 and number of edges connecting adjacent vertices is n. Thus, total number of edges is n*(n-1)/2.
Below is the implementation of above approach.
Minimum 0 Maximum 1
Time Complexity – O(n)
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
- Find K vertices in the graph which are connected to at least one of remaining vertices
- Count single node isolated sub-graphs in a disconnected graph
- Maximize the number of uncolored vertices appearing along the path from root vertex and the colored vertices
- Minimum number of edges between two vertices of a Graph
- Minimum number of edges between two vertices of a graph using DFS
- Finding in and out degrees of all vertices in a graph
- Number of Simple Graph with N Vertices and M Edges
- Queries to check if vertices X and Y are in the same Connected Component of an Undirected Graph
- Find if there is a path between two vertices in a directed graph
- Articulation Points (or Cut Vertices) in a Graph
- Check whether given degrees of vertices represent a Graph or Tree
- Largest subset of Graph vertices with edges of 2 or more colors
- Calculate number of nodes between two vertices in an acyclic Graph by Disjoint Union method
- Construct a graph from given degrees of all vertices
- Find if there is a path between two vertices in a directed graph | Set 2
- Find two disjoint good sets of vertices in a given graph
- Find if there is a path between two vertices in an undirected graph
- Minimize cost to color all the vertices of an Undirected Graph using given operation
- Check if every vertex triplet in graph contains two vertices connected to third vertex
- Calculate number of nodes between two vertices in an acyclic Graph by DFS method
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.