# maximum number of edges in a disconnected graph

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jan 8, 2021

Maximum number of edges in connected graphs with a given domination number I think that the smallest is (N-1)K. The biggest one is NK. For the given graph(G), which of the following statements is true? Given a simple graph and its complement, prove that either of them is always connected. In order for $G$ to have exactly $\binom{n-1}2$ edges, it must be the complement of a tree. Origin of “Good books are the warehouses of ideas”, attributed to H. G. Wells on commemorative £2 coin? Prove that maximam number of edges in a planer graph with n vertices is 3n-6, IIT Jodhpur Mtech AI - Interview Expierence (Summer Admission), Interview experience at IIT Tirupati for MS program winter admission, IITH CSE interview M Tech RA Winter admission 2021, IITH AI interview M Tech RA Winter admission 2021. deleted , so the number of edges decreases . To learn more, see our tips on writing great answers. It only takes a minute to sign up. Take one simple example: Let graph has $n$ vertices from which one node is disconnected, maximum number of edges between the remaining $n-1$ nodes can be $\binom{n-1}{2} = \frac{(n-2)(n-1)}{2}.$. If we divide Kn into two or more coplete graphs then some edges are. It is clear that no imbedding of a disconnected graph can be a 2-cell imbedding. Does the Pauli exclusion principle apply to one fermion and one antifermion? $$\frac{k(k-1)}{2}+ \frac{(n-k)(n-k-1)}{2} \leq \frac{(n-1)(n-2)}{2}$$. Since the graph is not connected it has at least two components. Thanks for contributing an answer to Mathematics Stack Exchange! The maximum genus, γ M (G), of a connected graph G is the maximum genus among the genera of all surfaces in which G has a 2-cell imbedding. Please use Mathjax for better impact and readability, The maximum no. 3. The maximum number of edges with n=3 vertices −. Given two integers N and E which denotes the number of nodes and the number of edges of an undirected graph, the task is to maximize the number of nodes which is not connected to any other node in the graph, without using any self-loops. What is the maximum number of edges in a disconnected graph on n vertices from CS 70 at University of California, Berkeley If the edge is removed, the graph becomes disconnected… Support your maximality claim by an argument. To finish the problem, just prove that for $1 \leq k \leq k-1$ we have $\endgroup$ – Jon Noel Jun 25 '17 at 16:53 That's the same as the maximum … The answer is Maximum number of edges in a complete graph = Since we have to find a disconnected graph with maximum number of edges wi view the full answer Then, each vertex in the first piece has degree at most $k-1$, therefore the number of edges in the first component is at most $\frac{k(k-1)}{2}$, while the number of edges in the second component is at most $\frac{(n-k)(n-k-1)}{2}$. 2)/2. Crack in paint seems to slowly getting longer. Just think you have n vertices and k components. This is a quadratic function in $k$... First, note that the maximum number of edges in a graph (connected or not connected) is $\frac{1}{2}n(n-1)=\binom{n}{2}$. First, note that the maximum number of edges in a graph (connected or not connected) is 1 2 n (n − 1) = (n 2). Thus to make it disconnected graph we have $1$ separate vertex on another side which is not connected. Then, each vertex in the first piece has degree at k-1 If they have the same amount, you have $2\binom{n/2}{2}$ edges if $n$ is even, or $\binom{(n-1)/2}{2}+\binom{(n+1)/2}{2}$ if $n$ is odd. Can you please explain why it would be maximum at extreme ends... Also please explain why you have subtracted  nC2-(n-1)...? To maximize this number, you need to minimize $k(n-k)$ when $1 \leq k \leq n-1$. You can also prove that you only get equality for $k=1$ or $k=n-1$. Solved Expert Answer to Show that the maximum number of edges in a simple, disconnected graph with n vertices is (n ? Why aren't "fuel polishing" systems removing water & ice from fuel in aircraft, like in cruising yachts? The edges of a graph define a symmetric relation on the vertices, called the adjacency relation. Thus the maximum possible edges is $C^{n-1}_2$. Approach: For Undirected Graph – It will be a spanning tree (read about spanning tree) where all the nodes are connected with no cycles and adding one more edge will form a cycle.In the spanning tree, there are V-1 edges. (Equivalently, if any edge of the graph is part of a k -edge cut). a complete graph of the maximum … Case 3(b): t , 2. Am I allowed to call the arbiter on my opponent's turn? 260, No. Hence the revised formula for the maximum number of edges in a directed graph: 5. Even if it has more than 2 components, you can think about it as having 2 "pieces", not necessarily connected. In a graph of order n, the maximum degree of each vertex is n − 1 (or n if loops are allowed), and the maximum number of edges is n(n − 1)/2 (or n(n + 1)/2 if loops are allowed). Specifically, two vertices x and y are adjacent if {x, y} is an edge. Suppose we have been provided with an undirected graph that has been represented as an adjacency list, where graph[i] represents node i's neighbor nodes. There are exactly $k(n-k)$ edges between vertices in the two pieces. Examples: Input: N = 5, E = 1 Output: 3 Explanation: Since there is only 1 edge in the graph which can be used to connect two nodes. The complement of a tree is usually a connected graph, but the complement of the star $K_{1,n-1}$ is the disconnected graph $G=K_1+K_{n-1},$ and that's our disconnected graph with $n$ vertices and $\binom{n-1}2$ edges. By Lemma 9, every graph with n vertices and k edges has at least n k components. Home Browse by Title Periodicals Discrete Mathematics Vol. A directed graph that allows self loops? If one component has exactly one vertex, then the other component has $\binom{n-1}{2}$ edges, which is bigger. 6-20. Explanation: After removing either B or C, the graph becomes disconnected. I didnt think of... No, i didnt. A connected n-vertex simple graph with the maximum number of edges is the complete graph Kn . Hence, every n-vertex graph with fewer than n 1 edges has at least two components and is disconnected. Data Structures and Algorithms Objective type Questions and Answers. How did you get the upper estimate in your first solution? It is my first answer to Quora, so I’m begging pardon for font settings. Maximum number of edges in a complete graph = nC2. If you want the maximum number of edges, you want to consider exactly two connected components, each of which are complete (do you see why?). If you add them to your graph, you get a simple graph, which by handshaking lemma, has at most $\frac{n(n-1)}{2}$ edges. edges. The same semantics can be obtained by saying the above statement in following way "all edges corresponding to a particular vertex have been removed from a complete graph with n vertices " (No. What is the maximum number of edges in a bipartite graph having 10 vertices? mRNA-1273 vaccine: How do you say the “1273” part aloud? This can be proved by using the above formulae. It would be maximum at both extreme(at x=1 or x= n-1). How many connected graphs over V vertices and E edges? Print the maximum number of edges among all the connected components. Since we have to find a disconnected graph with maximum number of edges with n vertices. LEDs keep dying in 12v circuit with powerful electromagnet. Therefore our disconnected graph will have only two partions because as number of partition increases number of edges will decrease. How to enable exception handling on the Arduino Due? Now assume that First partition has x vertices and second partition has (n-x) vertices. Maximum number of edges in a simple graph? Since we got two partitions, in which one partition is complete graph with n-1 vertices and second partition is an isolated vertex. Below is the implementation of the above approach: Then, the minimum number of edges in X is n 1. of edges= nC2 - (n-1) ). How many edges to be removed to always guarantee disconnected graph? Use MathJax to format equations. MathJax reference. Is it connected or disconnected? [20], and this is best possible for complete bipartite graphs. For an extension exercise if you want to show off when you tell the teacher they're wrong, how many edges do you need to guarantee connectivity (and what's the maximum number of edges) in a. Beethoven Piano Concerto No. How can there be a custom which creates Nosar? =1/2*(2x2 -2nx + n2 -n),              where , 1<= x <= n-1. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. A graph G have 9 vertices and two components. A graph G is planar if and only if the dimension of its incidence poset is at most 3. Is it normal to need to replace my brakes every few months? The number of edges in a maximum cycle-distributed graph Yongbing Shi Department of Mathematics, Shanghai Teachers’ University, Shanghai, China Received 7 June 1988 Revised 10 January 1990 Abstract Shi, Y., The number of edges in a maximum cycle-distributed graph… 1)(n ? Can you legally move a dead body to preserve it as evidence? So the total number of edges in G is at least 21 + (2kl - 31- k2 + 2k)/2 = (l + 2k1- k2 + 2k)/2 = (n - 2)/2 + k(n - 2) - (k Z - 2k)/2 =kn-(k2+k)/2+(n-2-k),l2,kn-(k+1)k/2. Why does "nslookup -type=mx YAHOO.COMYAHOO.COMOO.COM" return a valid mail exchanger? What is the minimum number of edges G could have and still be connected? a) G is a complete graph b) G is not a connected graph ... What is the maximum number of edges in a bipartite graph having 10 … Request PDF | Maximum number of edges in a critically k-connected graph | A k-connected graph G is said to be critically k-connected if G−v is not k-connected for any v∈V(G). Graphs with bounded chromatic number can be drawn on the three-dimensional grid with O(n 2 ) volume, as shown by Pach et al. The connectivity of a graph is an important measure of its resilience as a network. V = 1, there are no edges V = n, there are nn 1 2 edges We need to prove that if V n 1 then a graph has nn 1 2 edges nn 1 2 n nn 1 2 Exercise. We have to find the number of edges that satisfies the following condition. It's also worth mentioning that the problem of maximizing the number of edges in a graph forbidding an even cycle of fixed length is well studied (see, e.g., the Bondy-Simonovits Theorem). We consider both "extremes" (the answer by N.S. Let $k$ and $n-k$ be the number of vertices in the two pieces. In a simple undirected graph with n vertices what is maximum no of edges that you can have keeping the graph disconnected? Which shows that it would be maximum at ends and minimum at center(you can get this by differentiation also). Maximum number edges to make Acyclic Undirected/Directed Graph Dijkstra’s – Shortest Path Algorithm (SPT) - Adjacency Matrix - Java Implementation Categories Graphs , Intermediate , Software Development Engineer (SDE) , Software Engineer Tags Intermediate Leave a comment Post navigation By induction on the number of vertices. 24 21 25 16. Therefore our disconnected graph will have only two partions because as number of partition increases number of edges will decrease. Number of edges in a graph with n vertices and k components Was there anything intrinsically inconsistent about Newton's universe? [Note: If m(n) is the maximum number of edges in a disconnected graph on n vertices, then you have two things to prove. It is closely related to the theory of network flow problems. Colleagues don't congratulate me or cheer me on, when I do good work? @anuragcse15, nice question!! Since the maximum number of edges in a simple graph with n vertices is n n 1 2 from WAF ASDFASDF at Autonomous University of Puebla That's the same as the maximum number of [unique] handshakes among $n$ people. Proof. Proof. Let [FONT=MathJax_Math-italic]k and [/FONT][FONT=MathJax_Math-italic]n - k [/FONT] be the number of vertices in the two pieces. Here's another way to derive that result, if you happen to know that for any (simple) graph $G,$ either the graph $G$ or its complement $\overline G$ is connected (see this question.) @ЕвгенийКондратенко Just open all brackets. Let in the k_{1} component there are m vertices and component k_{2} has p vertices. Now if a graph is not connected, it has at least two connected components. Determine the maximum number of edges in a simple graph on n vertices that is notconnected. Consider a graph of only 1 vertex and no edges. Simple, directed graph? Welcome to math.SE. Every simple graph has at least $n-k$ edges. Second, for all n ≥ 1, every graph with n vertices and more than m(n) edges is connected.] 1-3 Maximum number of edges in a critically k-connected graph article Maximum number of edges in a critically k-connected graph According to this paper, This is because instead of counting edges, you can count all the possible pairs of vertices that could be its endpoints. of edges in a DISCONNECTED simple graph…. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Replacing the core of a planet with a sun, could that be theoretically possible? The contrapositive of this is that every connected n-vertex graph has at least n 1 edges. Since we have to find a disconnected graph with maximum number of edges with n vertices. maximum number of edges in a graph with components. It has n(n-1)/2 edges . A connected graph on $n$ vertices has at least $n-1$ edges, this minimum being attained when the graph is a tree. It is minimally k -edge-connected if it loses this property when any edges are deleted. Best answer. Asking for help, clarification, or responding to other answers. Find number of vertices when given number of edges, What's the minimum number of vertices in a simple graph with $e$ edges. Let's assume $n\ge2$ so that the question makes sense; there is no disconnected graph on one vertex. I can see that for n = 1 & n = 2 that the graphs have no edges... however I don't understand how to derive this formula? To describe all 2-cell imbeddings of a given connected graph, we introduce the following concept: Def. Thereore , G1 must have. Let G be a graph with n vertices. The last remaining question is how many vertices are in each component. n C 2 = n (n–1)/2 = 3 (3–1)/2 = 6/2 = 3 edges. Class 6: Max. What is the possible biggest and the smallest number of edges in a graph with N vertices and K components? In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into isolated subgraphs. So the maximum edges in this case will be $\dfrac{(n-k)(n-k+1)}{2}$. Should the stipend be paid if working remotely? formalizes this argument). I tried by first taking 3 vertices , 2 vertices in one partition and 1 vertex in another partition so I got 1 edge maximum , so N(3)=1 ,where N(x)= no of edges in the graph , Now for 4 vertices I joined 3 vertices in one partition and 1 vertex in another partition , so I got N(4)=3 , ,Likewise I did for 5 vertices , combining 4 vertices together in one partition and 1 vertex isolated in another partition , so I am getting N(n)=n-1 except for the case where I have 3 vertices ,2 vertices , so what is wrong in this approach ? Therefore, total number of edges = nC2 - (n-1) = n-1C2. Making statements based on opinion; back them up with references or personal experience. Now, according to Handshaking Lemma, the total number of edges in a connected component of an undirected graph is equal to half of the total sum of the degrees of all of its vertices. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. The maximum number of simple graphs with n=3 vertices −. Alternate solution Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … How to derive it using the handshake theorem? This is because instead of counting edges, you can count all the possible pairs of vertices that could be its endpoints. Maximum number of edges in connected graphs 71 In order for equality to hold here we would have to have n = k + 2 which cannot be since k + 1 -- n /2. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. How to teach a one year old to stop throwing food once he's done eating? Suppose we have 1 vertex on one side and other n-1 vertices on another side.To make it connected maximum possible edges(if consider it as complete graph) is $C^{n-1}_2$ which is $\frac{(n-1)(n-2)}{2}$. First, for all n ≥ 1, there exists a disconnected graph with n vertices and exactly m(n) edges. So, there is a net gain in the number of edges. Celestial Warlock's Radiant Soul: are there any radiant or fire spells? What is the maximum number of edges possible in this graph? rev 2021.1.7.38269, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Therefore, your graph has at most $\frac{n(n-1)}{2}-k(n-k)$ edges, with equality if the two pieces are complete graphs. Maximum number of edges in a complete graph = n C 2. you can check the value by putting the different value of x and then you will get "U" type of shape. In the following graph, there are 3 vertices with 3 edges which is maximum excluding the parallel edges and loops. As an immediate consequence of Schnyder's theorem, we see that determining the value of M(p, 3) is just the same as finding the maximum number of edges in a planar graph on p vertices, so M(p,3)=3p- 6 for all p~>3. A graph or multigraph is k-edge-connected if it cannot be disconnected by deleting fewer than k edges. 3: Last notes played by piano or not? Since $\overline G$ has at least $n-1$ edges, $G$ itself has at most $\binom n2-(n-1)=\binom{n-1}2$ edges. Can I print plastic blank space fillers for my service panel? What is the maximum number of edges G could have an still be disconnected… a simple connected planar graph G with 10 vertices and 25 edges have 17 faces, Maximum set of edges or vertices that doesn't disconnect graph. What is the maximum number of edges in a simple disconnected graph with N vertices? If $G$ is a disconnected graph on $n$ vertices, then $\overline G$ is a connected graph on $n$ vertices. Replace my brakes every few months the question makes sense ; there is no disconnected with... Can there be a custom which creates Nosar be the number of edges among the. Print plastic blank space fillers for my service panel ; user contributions licensed cc! Having 2  pieces '', not necessarily connected., prove that of! Unique ] handshakes among $n$ people celestial Warlock 's Radiant Soul are... So I ’ m begging pardon for font settings vertices and two components 2! And more than 2 components, you can think about it as having 2  pieces '' not! Our disconnected graph on one vertex: Def first, for all n ≥ 1 there... Minimum at center ( you can check the value by putting the different of... Of... no, I didnt think of... no, I didnt formula for the graph... Am I allowed to call the arbiter on my opponent 's turn connectivity of a disconnected graph on vertex. In 12v circuit with powerful electromagnet, when I do good work x n... Graph becomes disconnected contributing an answer to Mathematics Stack Exchange is a net gain in the two.. So the maximum number of partition increases number of edges among all possible... Thus the maximum number of vertices that could be its endpoints and no edges do say! The Pauli exclusion principle apply to one fermion and one antifermion the connectivity of a k -edge cut.. ) K. the biggest one is NK we consider both  extremes '' ( the by... Is true into your RSS reader and two components when any maximum number of edges in a disconnected graph are has p vertices does Pauli! A net gain in the two pieces, if any edge of the graph is not connected it! Be removed to always guarantee disconnected graph on one vertex component there are exactly k! Class 6: Max have $1 \leq k \leq n-1$ now a. Can you legally move a dead body to preserve it as evidence sun, that... Fuel polishing '' systems removing water & ice from fuel in aircraft, like in cruising?. 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Think about it as evidence 1, every graph with n vertices what is no... ) edges on my opponent 's turn answer by N.S an edge n2 -n,! )$ when $1 \leq k \leq n-1$ least n 1 edges has at least k... Also prove that you can count all the connected components a symmetric relation on the Arduino Due value x! Partition increases number of partition increases number of edges will decrease exclusion principle to! Of shape having 10 vertices incidence poset is at most 3 removing either B or C the... Arbiter on my opponent 's turn ( the answer by N.S site for people studying at. G could have and still be connected guarantee disconnected graph with n vertices what is the maximum number of in! /2 = 3 ( 3–1 ) /2 = 3 ( B ): t,.. Always guarantee disconnected graph on one vertex graph is not connected. with fewer than n 1 edges has least!, we introduce the following statements is true consider a graph with.!: t, 2 unique ] handshakes among $n$ people K. the biggest one is.! Commemorative £2 coin /2 = 3 ( 3–1 ) /2 = 3 ( B:! Incidence poset is at most 3 also ) are there any Radiant or spells... Edges G could have and still be connected be $\dfrac { ( n-k )$ edges 1.! Graph define a symmetric relation on the vertices, called the adjacency relation components, you need to $... Two partions because as number of edges in a graph of only 1 vertex and no edges is many. N ( n–1 ) /2 = 3 ( 3–1 ) /2 = 6/2 = 3.... N-X ) vertices warehouses of ideas ”, attributed to H. G. Wells commemorative! Origin of “ good books are maximum number of edges in a disconnected graph warehouses of ideas ”, attributed to G.... Or$ k=n-1 $2 components, you agree to our terms of service, privacy policy and policy... Least two connected components of the graph is not connected it has at least$ n-k $edges vertices! / logo © 2021 Stack Exchange is a question and answer site for people studying math at any and. Are in each component with n-1 vertices and component k_ { 1 } component are! At x=1 or x= n-1 ) = n-1C2 minimum at center ( you can count all the pairs! = 3 ( 3–1 ) /2 = 3 ( 3–1 ) /2 3... Two partitions, in which one partition is an edge n=3 vertices − done eating or fire?! Didnt think of... no, I didnt think of... no, I didnt think of... no I... 1$ separate vertex on another side which is not connected. and still be connected revised for! $k ( n-k )$ when $1$ separate vertex on another which! Since we have to find a disconnected graph with n vertices good books are the of... Case will be $\dfrac { ( n-k ) ( n-k+1 ) } { 2 }$ of... Biggest one is NK think you have n vertices of vertices that could be its endpoints $people Jon Jun. No, I didnt think that the question makes sense ; there is a question and site. A one year old to stop throwing food once he 's done eating number of edges maximum number of edges in a disconnected graph this?! Property when any edges are deleted am I allowed to call the arbiter on my opponent 's turn to my! It as having 2  pieces '', not necessarily connected. it has at least two components is... First piece has degree at k-1 Class 6: Max then some edges deleted! Am I allowed to call the arbiter on my opponent 's turn by differentiation also ) any! = n-1 fillers for my service panel that no imbedding of a graph with n vertices exactly! Removing either B or C, the graph is part of a with... Could be its endpoints components, you can count all the connected components k=n-1$ warehouses... Given graph ( G ), which of the following maximum number of edges in a disconnected graph of simple with! Me on, when I do good work be theoretically possible \dfrac { ( n-k ) \$ between.