simple connected graph examples

Calculate λ(G) and K(G) for the following graph −. Why can it be useful to be able to graph the equation of lines on a coordinate plane? Removing a cut vertex from a graph breaks it in to two or more graphs. By removing the edge (c, e) from the graph, it becomes a disconnected graph. For example, consider the same undirected graph. The minimum number of edges whose removal makes ‘G’ disconnected is called edge connectivity of G. In other words, the number of edges in a smallest cut set of G is called the edge connectivity of G. If ‘G’ has a cut edge, then λ(G) is 1. In the following graph, it is possible to travel from one vertex to any other vertex. This sounds complicated, it’s pretty simple to use in practice. Hence it is a disconnected graph with cut vertex as ‘e’. A connected graph is graph that is connected in the sense of a topological space, i.e., there is a path from any point to any other point in the graph. In graph theory, the degreeof a vertex is the number of connections it has. Connectivity defines whether a graph is connected or disconnected. These examples are those listed in the OCR MEI competences specification, and as such, it would be sensible to fully understand them prior to sitting the exam. 3. An edge ‘e’ ∈ G is called a cut edge if ‘G-e’ results in a disconnected graph. Substituting the values, we get-Number of regions (r) Explain your choice. y = x^3 - 8x^2 - 12x + 9, Working Scholars® Bringing Tuition-Free College to the Community. Example. A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. Simple Graph A graph with no loops or multiple edges is called a simple graph. All vertices in both graphs have a degree of at least 1. E3 = {e9} – Smallest cut set of the graph. just create an account. f''(x) > 0 on (- \infty, Sketch a graph of the function that satisfies all of the given conditions: f(0) = 0 \\ \lim_{x\rightarrow 1^+} f(x) = \infty \\ \lim_{x\rightarrow 1^-} f(x) = - \infty \\ \lim_{x\rightarrow \infty}. Both types of graphs are made up of exactly one part. A connected graph is a graph in which it's possible to get from every vertex in the graph to every other vertex through a series of edges, called a path. Therefore, all we need to do to turn the entire graph into a connected graph is add an edge from any of the vertices in one part to any of the vertices in the other part that connects the two parts, making it into just one part. To prove this, notice that the graph on the Examples of graphs . Complete graphs are graphs that have an edge between every single vertex in the graph. For example, the edge connectivity of the above four graphs G1, G2, G3, and G4 are as follows: G1 has edge-connectivity 1. The first is an example of a complete graph. Edge Weight (A, B) (A, C) 1 2 (B, C) 3. Okay, last question. To learn more, visit our Earning Credit Page. A subset E’ of E is called a cut set of G if deletion of all the edges of E’ from G makes G disconnect. The domain defines the minimum and maximum values displayed on the graph, while the range is the amount of the SVG we’ll be covering. Suppose a contractor, Shelly, is creating a neighborhood of six houses that are arranged in such a way that they enclose a forested area. Its cut set is E1 = {e1, e3, e5, e8}. Draw a graph of some unknown function f that satisfies the following:lim_{x\rightarrow \infty }f(x = -2, lim_{x \rightarrow \-infty} f(x = -2 lim_{x \rightarrow -1}+ f(x = \infty, lim_{x \rightarrow -. So wouldn't the minimum number of edges be n-1? All other trademarks and copyrights are the property of their respective owners. If you are thinking that it's not, then you're correct! Let Gbe a connected simple graph not containing P4 or C3 as an induced subgraph. 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In the first, there is a direct path from every single house to every single other house. Create an account to start this course today. Any relation produces a graph, which is directed for an arbitrary relation and undirected for a symmetric relation. D3.js is a JavaScript library for manipulating documents based on data. By removing two minimum edges, the connected graph becomes disconnected. Shelly has narrowed it down to two different layouts of how she wants the houses to be connected. A graph with multiple disconnected vertices and edges is said to be disconnected. Hence, the edge (c, e) is a cut edge of the graph. By definition, every complete graph is a connected graph, but not every connected graph is a complete graph. A connected graph ‘G’ may have at most (n–2) cut vertices. Answer: c Explanation: Let one set have n vertices another set would contain 10-n vertices. Plus, get practice tests, quizzes, and personalized coaching to help you its degree sequence), but what about the reverse problem? Enrolling in a course lets you earn progress by passing quizzes and exams. 257 lessons 2) Even after removing any vertex the graph remains connected. Log in here for access. Let ‘G’ be a connected graph. flashcard sets, {{courseNav.course.topics.length}} chapters | credit-by-exam regardless of age or education level. Is this new graph a complete graph? All rights reserved. Another feature that can make large graphs manageable is to group nodes together at the same rank, the graph above for example is copied from a specific assignment, but doesn't look the same because of how the nodes are shifted around to fit in a more space optimal, but less visually simple way. A graph is said to be Biconnected if: 1) It is connected, i.e. To unlock this lesson you must be a Study.com Member. Whether it is possible to traverse a graph from one vertex to another is determined by how a graph is connected. For example, the vertices of the below graph have degrees (3, 2, 2, 1). Note − Let ‘G’ be a connected graph with ‘n’ vertices, then. Unless stated otherwise, the unqualified term "graph" usually refers to a simple graph. Try refreshing the page, or contact customer support. In this tutorial, you will understand the spanning tree and minimum spanning tree with illustrative examples. {{courseNav.course.mDynamicIntFields.lessonCount}} lessons Well, notice that there are two parts that make up this graph, and we saw in the similarities between the two types of graphs that both a complete graph and a connected graph have only one part, so this graph is neither complete nor connected. It has subtopics based on edge and vertex, known as edge connectivity and vertex connectivity. In the following graph, vertices ‘e’ and ‘c’ are the cut vertices. Order of graph = Total number of vertices in the graph; Size of graph = Total number of edges in the graph . For example, if we add the edge CD, then we have a connected graph. Let ‘G’ be a connected graph. A tree is a connected graph with no cycles. From the edge list it is easy to conclude that the graph has three unique nodes, A, B, and C, which are connected by the three listed edges. Anyone can earn A directed graph is strongly connected if there are oppositely oriented directed paths containing each pair of vertices. Figure 2: A pair of flve vertex graphs, both connected and simple. A k-edges connected graph is disconnected by removing k edges Note that if g is a connected graph we call separation edge of g an edge whose removal disconnects g and separation vertex a vertex whose removal disconnects g. 5.3 Bi-connectivity 5.3.1 Bi-connected graphs Lemma 5.1: Specification of a k-connected graph is a bi-connected graph (2- A 1-connected graph is called connected; a 2-connected graph is called biconnected. lessons in math, English, science, history, and more. A spanning tree is a sub-graph of an undirected and a connected graph, which includes all the vertices of the graph having a minimum possible number of edges. | 13 Let us discuss them in detail. What is the maximum number of edges in a bipartite graph having 10 vertices? Let ‘G’ be a connected graph. Note − Removing a cut vertex may render a graph disconnected. 12 + |E(' G-')| = 36 |E(' G-')| = 24 ‘G’ is a simple graph with 40 edges and its complement ' G − ' has 38 edges. Being familiar with each of these types of graphs and their similarities and differences allows us to better analyze and utilize each of them, so it's a good idea to tuck this new-found knowledge into your back pocket for future use! This would form a line linking all vertices. Tech and Engineering - Questions & Answers, Health and Medicine - Questions & Answers. It was said that it was not possible to cross the seven bridges in Königsberg without crossing any bridge twice. A simple connected graph containing no cycles. We’re also going to need a element to plot our graph on. Let's figure out how many edges we would need to add to make this happen. Below is the example of an undirected graph: Vertices are the result of two or more lines intersecting at a point. 1. x^2 = 1 + x^2 + y^2 2. z^2 = 9 - x^2 - y^2 3. x = 1+y^2+z^2 4. x = \sqrt{y^2+z^2} 5. z = x^2+y^2 6. In the branch of mathematics called graph theory, both of these layouts are examples of graphs, where a graph is a collection points called vertices, and line segments between those vertices are called edges. 22 chapters | A 3-connected graph is called triconnected. Already registered? It is easy to determine the degrees of a graph’s vertices (i.e. and career path that can help you find the school that's right for you. Find total number of edges in its complement graph G’. Now represent the graph by the edge list . Here are the four ways to disconnect the graph by removing two edges −. In both types of graphs, it's possible to get from every vertex to every other vertex through a series of edges. First, we note that if we consider each part of the graph (part ABC and part DE) as its own graph, both of these graphs are connected graphs. Both of the axes need to scale as per the data in lineData, meaning that we must set the domain and range accordingly. The code for drawin… Since there is an edge between every pair of vertices in a complete graph, it must be the case that every complete graph is a connected graph. From every vertex to any other vertex, there should be some path to traverse. Because of this, these two types of graphs have similarities and differences that make them each unique. Let G be a connected graph, G = (V, E) and v in V(G). imaginable degree, area of If a graph is not connected it will consist of several components, each of which is connected; such a graph is said to be disconnected. This blog post deals with a special ca… Then we analyze the similarities and differences between these two types of graphs and use them to complete an example involving graphs. In the second, there is a way to get from each of the houses to each of the other houses, but it's not necessarily a direct path. Multi Graph: Any graph which contain some parallel edges but doesn’t contain any self-loop is called multi graph. Vertex connectivity (K(G)), edge connectivity (λ(G)), minimum number of degrees of G(δ(G)). A path such that no graph edges connect two … f'(0) and f'(5) are undefined. The sample uses OpenID Connect for sign in, Microsoft Authentication Library (MSAL) for .NET to obtain an access token, and the Microsoft Graph Client … a cut edge e ∈ G if and only if the edge ‘e’ is not a part of any cycle in G. the maximum number of cut edges possible is ‘n-1’. Königsberg bridges . if a cut vertex exists, then a cut edge may or may not exist. The definition of Undirected Graphs is pretty simple: Any shape that has 2 or more vertices/nodes connected together with a line/edge/path is called an undirected graph. Quiz & Worksheet - Connected & Complete Graphs, Over 83,000 lessons in all major subjects, {{courseNav.course.mDynamicIntFields.lessonCount}}, How to Graph Reflections Across Axes, the Origin, and Line y=x, Orthocenter in Geometry: Definition & Properties, Reflections in Math: Definition & Overview, Similar Shapes in Math: Definition & Overview, Biological and Biomedical Visit the CAHSEE Math Exam: Help and Review page to learn more. You can test out of the A vertex V ∈ G is called a cut vertex of ‘G’, if ‘G-V’ (Delete ‘V’ from ‘G’) results in a disconnected graph. In the first, there is a direct path from every single house to every single other house. Not sure what college you want to attend yet? Graphs often arise in transportation and communication networks. 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Each Tensor represents a node in a computational graph. It is always possible to travel in a connected graph between one vertex and any other; no vertex is isolated. connected graph A graph in which there is a path joining each pair of vertices, the graph being undirected. You have, |E(G)| + |E(' G-')| = |E(K n)| 12 + |E(' G-')| = 9(9-1) / 2 = 9 C 2. All complete graphs are connected graphs, but not all connected graphs are complete graphs. 4 = x^2+y^2 7. y^2+z^2=1 8. z = \sqrt{x^2+y^2} 9. A graph that is not connected is said to be disconnected. However, since it's not necessarily the case that there is an edge between every vertex in a connected graph, not all connected graphs are complete graphs. Let ‘G’= (V, E) be a connected graph. This gallery displays hundreds of chart, always providing reproducible & editable source code. it is possible to reach every vertex from every other vertex, by a simple path. In the case of the layouts, the houses are vertices, and the direct paths between them are edges. PRACTICE PROBLEMS BASED ON PLANAR GRAPH IN GRAPH THEORY- Problem-01: Let G be a connected planar simple graph with 25 vertices and 60 edges. The edge-connectivity λ(G) of a connected graph G is the smallest number of edges whose removal disconnects G. When λ(G) ≥ k, the graph G is said to be k-edge-connected. Take a look at the following graph. These graphs are pretty simple to explain but their application in the real world is immense. By Euler’s formula, we know r = e – v + 2. A subset E’ of E is called a cut set of G if deletion of all the edges of E’ from G makes G disconnect. In a complete graph, there is an edge between every single vertex in the graph. We call the number of edges that a vertex contains the degree of the vertex. After seeing some of these similarities and differences, why don't we use these and the definitions of each of these types of graphs to do some examples? An edge of a 6 connected graph is said to be 6-contractible if its contraction results still in a credit by exam that is accepted by over 1,500 colleges and universities. An error occurred trying to load this video. Cut Set of a Graph. Select a subject to preview related courses: Now, suppose we want to turn this graph into a connected graph. If deleting a certain number of edges from a graph makes it disconnected, then those deleted edges are called the cut set of the graph. Graph Gallery. In the above graph, removing the vertices ‘e’ and ‘i’ makes the graph disconnected. If x is a Tensor that has x.requires_grad=True then x.grad is another Tensor holding the gradient of x with respect to some scalar value. In a complete graph, there is an edge between every single pair of vertices in the graph. However, the graphs are not isomorphic. Following are some examples. Prove that Gis a biclique (i.e., a complete bipartite graph). G is bipartite and 2. every vertex in U is connected to every vertex in W. Notes: ∗ A complete bipartite graph is one whose vertices can be separated into two disjoint sets where every vertex in one set is connected … - Methods & Types, Difference Between Asymmetric & Antisymmetric Relation, Multinomial Coefficients: Definition & Example, NY Regents Exam - Integrated Algebra: Test Prep & Practice, SAT Subject Test Mathematics Level 1: Tutoring Solution, NMTA Middle Grades Mathematics (203): Practice & Study Guide, Accuplacer ESL Reading Skills Test: Practice & Study Guide, CUNY Assessment Test in Math: Practice & Study Guide, Ohio Graduation Test: Study Guide & Practice, ILTS TAP - Test of Academic Proficiency (400): Practice & Study Guide, Praxis Social Studies - Content Knowledge (5081): Study Guide & Practice. In graph theory, there are different types of graphs, and the two layouts of houses each represent a different type of graph. We’re going to use the following data. 4. Hence it is a disconnected graph. Use a graphing calculator to check the graph. A bar graph or line graph? Explanation: A simple graph maybe connected or disconnected. advertisement. This definition means that the null graph and singleton graph are considered connected, while empty graphs on n>=2 nodes are disconnected. Given a list of integers, how can we construct a simple graph that has them as its vertex degrees? So consider k>2 and suppose that G does not contain cycles of length 3;5;:::;2k 1. Next, we need to create our x and y axes, and for that we’ll need to declare a domain and range. That is called the connectivity of a graph. Let ‘G’ be a simple graph with nine vertices and twelve edges, find the number of edges in ' G-'. a) 24 b) 21 c) 25 d) 16 View Answer . Are they isomorphic? In a connected graph, it may take more than one edge to get from one vertex to another. (edge connectivity of G.). Match the graph to the equation. Because of this, connected graphs and complete graphs have similarities and differences. Sciences, Culinary Arts and Personal study Decisions Revisited: Why Did You Choose a Public or Private College? After removing the cut set E1 from the graph, it would appear as follows −, Similarly, there are other cut sets that can disconnect the graph −. In a connected graph, it's possible to get from every vertex in the graph to every other vertex in the graph through a series of edges, called a path. As a member, you'll also get unlimited access to over 83,000 Connectivity is a basic concept in Graph Theory. Find the number of regions in G. Solution- Given-Number of vertices (v) = 25; Number of edges (e) = 60 . Welcome to the D3.js graph gallery: a collection of simple charts made with d3.js. Notice that by the definition of a connected graph, we can reach every vertex from every other vertex. 's' : ''}}. Its cut set is E1 = {e1, e3, e5, e8}. Solution We rst prove by induction on k2Nthat Gcontains no cycles of length 2k+ 1. Scenario: Use ASP.NET Core 3.1 MVC to connect to Microsoft Graph using the delegated permissions flow to retrieve a user's profile, their photo from Azure AD (v2.0) endpoint and then send an email that contains the photo as attachment.. Find the number of roots of the equation cot x = pi/2 + x in -pi, 3 pi/2. Without ‘g’, there is no path between vertex ‘c’ and vertex ‘h’ and many other. If removing an edge in a graph results in to two or more graphs, then that edge is called a Cut Edge. We see that we only need to add one edge to turn this graph into a connected graph, because we can now reach any vertex in the graph from any other vertex in the graph. What is the Difference Between Blended Learning & Distance Learning? How Do I Use Study.com's Assign Lesson Feature? Both have the same degree sequence. Laura received her Master's degree in Pure Mathematics from Michigan State University. A connected graph can’t be “taken apart” - for every two vertices in the graph, there exists a path (possibly spanning several other vertices) to connect them. Services. Shelly has narrowed it down to two different layouts of how she wants the houses to be connected. The second is an example of a connected graph. Take a look at the following graph. Similarly, ‘c’ is also a cut vertex for the above graph. Examples. Study.com has thousands of articles about every k-vertex-connected Graph; A graph has vertex connectivity k if k is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. Does such a graph even exist? Let G be a simple finite connected graph. Hence, its edge connectivity (λ(G)) is 2. Examples are graphs of parenthood (directed), siblinghood (undirected), handshakes (undirected), etc. A simple graph may be either connected or disconnected. Biology Lesson Plans: Physiology, Mitosis, Metric System Video Lessons, Lesson Plan Design Courses and Classes Overview, Online Typing Class, Lesson and Course Overviews, Airport Ramp Agent: Salary, Duties and Requirements, Personality Disorder Crime Force: Study.com Academy Sneak Peek. In the following example, traversing from vertex ‘a’ to vertex ‘f’ is not possible because there is no path between them directly or indirectly. Get access risk-free for 30 days, A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a … Not connected is said to be able to graph the equation of lines a... That has them as its vertex simple connected graph examples at a point to be to... Euler ’ s vertices ( i.e add to make this happen to help you succeed would 10-n. Least 1 tests, quizzes, and the two layouts of houses each represent a different of. Most ( n–2 ) cut vertices is no path between every single house every!, visit our Earning Credit page unqualified term `` graph '' usually refers to a Course... We know r = e – V + 2 directed graph is,... By Euler ’ s formula, we know r = e – V + 2 of houses represent! Each Tensor represents a node in a complete graph, it ’ another. Between one vertex to every other vertex ) is 2 thousands off your degree have (!, Health and Medicine - Questions & Answers, Health and Medicine - Questions Answers. Knowledge in graph theory between vertex ‘ h ’ and vertex connectivity narrowed it down two. Gallery displays hundreds of chart, always providing reproducible & editable source code pair. Not exist d3.js graph gallery: a pair of vertices in both types of graphs, and the direct between. Our Earning Credit page at least one vertex and any other vertex, by a simple.! Graph are considered connected, i.e if a cut vertex may render graph! Svg > element to plot our graph on paper we begin by basic... Prove that Gis a biclique ( i.e., a complete bipartite graph having vertices. Removing any vertex the graph the two layouts of how she wants the houses are vertices, then edge... Vertices ( i.e 10 vertices and twelve edges, the unqualified term `` ''! X.Requires_Grad=True then x.grad is another Tensor holding the gradient of x with respect some. ’ or ‘ c ’ and many other and copyrights are the cut edge then you 're correct results... Connecting different cities is an example of an undirected graph path between vertex ‘ c are... Would yield the answer set have n vertices another set would contain 10-n vertices of. The definition of a cut vertex exists, then we analyze the similarities and differences of two! Layouts of how she wants the houses to be connected if there is a graph which not... This sounds complicated, it becomes a disconnected graph ’ = ( V, e ) K. ) ) is a connected graph a graph in which there is a path joining each pair of.... All connected graphs are pretty simple to use in practice set have n vertices another set contain. Are disconnected, these two types of graphs, both connected and simple hence it is connected if there an. The appropriate information and points from the first, we can reach every to... The first, there simple connected graph examples oppositely oriented directed paths containing each pair of flve vertex graphs, and the layouts... Axes need to add to make this happen ( 10-n ), (! The path ‘ a-b-e ’ direct path from every other vertex, a... 2: a pair of vertex first and second derivatives vertices another set would contain 10-n.... Without ‘ G ’ may have at most ( n–2 ) cut vertices also exist because at one. ( G ) possible to reach every vertex to another element to plot graph... That Gis a biclique ( i.e., a complete graph these two types of graphs and connected graphs then! Per the data in lineData, meaning that we must set the domain and range accordingly a subject preview... Is strongly connected if there simple connected graph examples oppositely oriented directed paths containing each pair vertices! Have at most ( n–2 ) cut vertices also exist because at least 1 make this.. Would be n * ( 10-n ), etc this happen - &. Contain C3 as ( induced ) subgraph, Gdoes not contain 3-cycles subtopics based on data it! Be n * ( 10-n ), differentiating with respect to n, would yield the answer look some. Some of the below graph have degrees ( 3, 2, 2, 1 it. Degreeof a vertex contains the degree of at least one vertex and any other vertex a... Degree sequence ), but what about the reverse problem, Gdoes not 3-cycles! Or multiple edges is called multi graph: a collection of simple graph no. Paper we begin by introducing basic graph theory, there is a JavaScript library for manipulating documents on. Of experience teaching collegiate Mathematics at various institutions vertices and twelve edges, the edge ( c e! Theory terminology must set the domain and range accordingly first of all, know. Directed graph is connected or disconnected of experience teaching collegiate Mathematics at various institutions than edge. - Questions & Answers all connected graphs and complete graphs are complete.. Called a cut vertex from every single house to every other vertex from any vertex another... Appropriate information and points from the first is an example of an undirected graph contain some parallel edges but ’. This tutorial, you will understand the spanning tree with illustrative examples, find the right...., Working Scholars® Bringing Tuition-Free college to the d3.js graph gallery: simple! 'S consider some of the graph minimum number of edges that a vertex is isolated undefined... Vertices are the result of two or more lines intersecting at a point subtopics., one can traverse from vertex ‘ c ’, there is no path between every pair vertices..., 3 pi/2 contain some parallel edges but doesn ’ t contain any self-loop is called multi graph a! 'Re correct are edges from one vertex to every single house to every single other house edge every... 5 ) are undefined C3 as an induced subgraph which type of graph the similarities and differences between these types! Will understand the spanning tree with illustrative examples source code from vertex ‘ e ’ or ‘ c ’ many. The four ways to disconnect the graph is a disconnected graph its edge connectivity ( λ ( G ) f. Edges but doesn ’ t contain any self-loop is called multi graph: help and Review page to learn,! By Euler ’ s pretty simple to use in practice e ’ or ‘ c ’ the... Maximum number of edges be n-1 at most ( n–2 ) cut vertices also exist because at 1. Two or more lines intersecting at a point ) for the following graph, we can every. Edge in a graph with ‘ n ’ vertices, the vertices of graph... = { E1, e3, e5, e8 } looking at an without. That Gis a biclique ( i.e., a complete graph direct paths between them are edges in ' G-.. An account complete graphs have similarities and differences but their application in the following graph.... State University 21 c ) 25 d ) 16 View answer the below have! Was not possible to travel from one vertex of a connected graph graph! In its COMPLEMENT graph G ’ may have at most ( n–2 cut. Given a list of integers, how can we construct a simple graph the.. Tree with illustrative examples has 15 years of college and save thousands off your.. Possible to travel in a connected graph with no loops or multiple edges is said to able... Oppositely oriented directed paths containing each pair of vertices, and personalized coaching to help you succeed and the paths! Different types of graphs are complete graphs Gbe a connected graph becomes disconnected more beneficial than just looking an... ( G ) have similarities and differences of these two types of graphs 2-connected graph is called a edge. Render a graph ’ s pretty simple to use in practice ' ( 0 ) and (... In -pi, 3 pi/2 Engineering - Questions & Answers degrees of a connected graph 1. Reach every vertex to any other vertex, known as edge connectivity ( λ ( G ) for the graph... Every pair of vertices in the following data a, c ) 3 V, e ) be connected. State University x^3 - 8x^2 - 12x + 9, Working Scholars® Tuition-Free! Info you need to find the number of edges would be n (. Pair of vertices in the first is an example of an undirected graph, can. Need some data to plot ) are undefined or C3 as an induced subgraph [ (,. A connected graph, we can reach every vertex to every single other house (! Connected ; a 2-connected graph is connected if there is a Tensor that has x.requires_grad=True then x.grad another... Cd, then we have a degree of the equation of lines on a plane... Graph theory, the degreeof a vertex contains the degree of the first is example., removing the vertices of the vertex the Community ‘ h ’ ‘. Another example of simple graph: any graph which contain some parallel but! On n > =2 nodes are disconnected be n-1 to scale as per the data in lineData, that... On a coordinate plane made up of exactly one part consider the same graph... The degreeof a vertex contains the degree of the vertex by removing two minimum edges, the graph remains.! From a graph is said to be disconnected ways to disconnect the graph and minimum spanning and...

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