Since your post mentions explicitly bipartite graphs and adjacency matrix, here is a possibility. Draw a Bipartite graph? d) disjoint vertex set We provide all important questions and answers from chapter Discrete Mathematics. b) There is no beginning and no end. b) 2-vertex set of G1 Next Page Previous Page Discrete Mathematics MCQs for Software Engineering Students 1. The complete bipartite graph K m, n is planar if and only if m ≤ 2 or n ≤ 2. I was going through Modern Graph Theory by Bollobas and this example on bipartite graph which states on the last line of page 6 that Figure I.1 is a bipartite graph. A graph is an ordered pair G = (V, E) comprising a set V of vertices or nodes and a collection of pairs of vertices from V called edges of the graph. This is, of course, the cannon with which to shoot a mosquito. A bipartite graph that doesn't have a matching might still have a partial matching. d) 87 Please visit using a browser with javascript enabled. 4 Add an edge from every vertex in B to t. 5 Make all the capacities 1. b) linear time 1. What is meant by Two-colorable graph? Show that if every component of a graph is bipartite, then the graph is bipartite. C) r=(r+1)% QUEUE_SIZE 13. The time complexity to test whether a graph is bipartite or not is said to be _______ using depth first search. A complete bipartite graph is a one in which each vertex in set X has an edge with set Y. These quiz objective questions are helpful for competitive exams. The simple non-planar graph with minimum number of edges is K 3, 3. Sanfoundry Global Education & Learning Series – Discrete Mathematics. Page-5 It is not difficult to prove that a graph is bipartite if and only if it does not have a cycle of an odd length. What is meant by Bipartite graph? So bipartite graphs belongs to class 1 graphs. c) odd b) 14 Let n be the total number of vertices. Example. We can prove it in this following way. If the number of branches in a network is B, the number of nodes is N, the number of independent loops is L, then the number of independent node equations will be N + L – 1 B – 1 N – 1 B – N 2. What is the relation between them? Jan 02,2021 - Graphs Theory MCQ - 1 | 20 Questions MCQ Test has questions of Computer Science Engineering (CSE) preparation. 5. We can also say that there is no edge that connects vertices of same set. b) null Polyhedral graph A Bipartite Graph is a graph whose vertices can be divided into two independent sets, U and V such that every edge (u, v) either connects a vertex from U to V or a vertex from V to U. A graph is bipartite iff its vertices can be divided into two sets, such that every edge connects a vertex from set 1 to one in set 2. If it can be divided into two independent sets A and B such that each edge connects a vertex from to A to B, If the graph is connected and it has odd number of vertices, If the graph has at least n/2 vertices whose degree is greater than n/2. d) reflexive, planar Please wait while the activity loads. Since the given edge adds exactly once to both U and V we can tell that this statement is true for all n vertices. Data Structure MCQ - Graph. Nevertheless, as @Dal said in comments, this is far from being the only solution; there is no silver bullet when it comes to representing graphs. For maximum number of edges, the total number of vertices hat should be present on set X is? D) Traversal 17. a) bipartition of G1 A complete graph K n is planar if and only if n ≤ 4. Let’s consider a graph .The graph is a bipartite graph if:. Null Graph. Thus K 4 is a planar graph. Let n be the total number of vertices. d) O(nlogn) Let x be the total number of vertices on set X. This section focuses on the "Graph" of the Data Structure. 1. Given that the bipartitions of this graph are U and V respectively. In the above graph, there are … Graph Theory Multiple Choice Questions and Answers for competitive exams. If loading fails, click here to try again. The partition V = V1 ∪ V2 in a bipartite graph G1 is called ________ In other words, for every edge (u, v), either u belongs to U and v to V, or u belongs to V and v to U. Every bipartite graph is 2 – chromatic. A k-coloring of G is an assignment of k colors to the vertices of G in such a way that adjacent vertices are assigned different colors. This test is Rated positive by 92% students preparing for Computer Science Engineering (CSE).This MCQ test is related to Computer Science Engineering (CSE) syllabus, prepared by Computer Science Engineering (CSE) teachers. Fig I.1 has triangles as ... graph-theory bipartite-graphs We know that for a connected planar graph 3v-e≥6.Hence for K 4, we have 3x4-6=6 which satisfies the property (3). B) Bipartite 11. 4. This section focuses on "Graph" in Discrete Mathematics. Solved MCQ on Tree and Graph in Data Structure set-1 ... 10. Properties of Bipartite Graphs Multiple choice Questions and Answers (MCQs). Gkseries provide you the detailed solutions on Discrete Mathematics as per exam pattern, to help you in day to day learning. In other words, for every edge (u, v), either u belongs to U and v to V, or u belongs to V and v to U. View Answer, 10. c) 210 If you leave this page, your progress will be lost. 7. as you can see you have created two partite sets, therefore this is a bipartite graph answered Aug 15, 2020 by Arkaprava ( 717 points) edited Aug 15, 2020 by Arkaprava A simple non-planar graph with minimum number of vertices is the complete graph K 5. A k-regular bipartite graph is the one in which degree of each vertices is k for all the vertices in the graph. In contrast, bipartite graphs contain too many 2-cliques and 2-clans. Given G is a bipartite graph and the bipartitions of this graphs are U and V respectively. MCQ 71: A graph is said to be _____ if the vertices can be split into two sets V1 and V2 such there are no edges between two vertices of V1 or two vertices of V2. Vertex Coloring. There are four students in a class namely A, B, C and D. A tells that a triangle is a bipartite graph. 2. Prove that if uis a vertex of odd degree in a graph, then there exists a path from uto another By adding one edge, the degree of vertices in U is equal to 1 as well as in V. Let us assume that this is true for n-1 edges and add one more edge. 4) A graph with no edges is known as empty graph. We go over it in today’s lesson! Maximum Cardinality Bipartite Matching (MCBM) Bipartite Matching is a set of edges \(M\) such that for every edge \(e_1 \in M\) with two endpoints \(u, v\) there is no other edge \(e_2 \in M\) with any of the endpoints \(u, v\). View Answer, 5. Bipartite graph A bipartite graph, also called a bigraph, is a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacentExamples: Regular graph. These Multiple Choice Questions (mcq) should be practiced to improve the Data Structure skills required for various interviews (campus interview, walk-in interview, company interview), placement, entrance exam and other competitive examinations. 3. Congratulations - you have completed Bipartite Graph Multiple choice Questions and Answers (MCQs). An electric circuit with 10 branches and ... -graph-theory-mcq/" aria-label="More on Network Graph Theory MCQ">Read more This test is Rated positive by 92% students preparing for Computer Science Engineering (CSE).This MCQ test is related to Computer Science Engineering (CSE) syllabus, prepared by Computer Science Engineering (CSE) teachers. Here's one that is very relevant to e-commerce, which touches our daily lives: We can model customer purchases of products using a bipartite graph. What is the maximum number of edges in a bipartite graph on 14 vertices? In a complete bipartite graph, the intersection of two sub graphs is ______ © 2011-2020 Sanfoundry. Bipartite Matching is a set of edges M M such that for every edge e1 ∈ M e 1 ∈ M with two endpoints u,v u, v there is no other edge e2 ∈ M e 2 ∈ M with any of the endpoints u,v u, v. A matching is said to be maximum if there is no other matching with more edges. This set of MCQ questions on data structure includes solved objective questions on graph, tree, and tree traversal. Get to the point NTA-NET (Based on NTA-UGC) Computer Science (Paper-II) questions for your exams. If G has a k-coloring, then G is said to be k-coloring, then G is said to be k-colorable.The chromatic number of G, denoted by X(G), is the smallest number k for which is k-colorable. MCQ 72: In a queue, the initial values of front pointer f rare pointer r should be _____ and _____ respectively. Bipartite graph belongs to class 1 graphs. Every bipartite graph (with at least one edge) has a partial matching, so we can look for the largest partial matching in a graph. 5. Another possibility is to transform my directed graph into an undirected one simply by adding the missing edges (e.g., if there is an edge from node … b) point graph A graph is bipartite if it is 2-colourable. Using Net Flow to Solve Bipartite Matching To Recap: 1 Given bipartite graph G = (A [B;E), direct the edges from A to B. C) 4 16. The maximum number of edges in a bipartite graph on 14 vertices is ___________ d) subgraph 2 Add new vertices s and t. 3 Add an edge from s to every vertex in A. View Answer, 3. These short objective type questions with answers are very important for Board exams as well as competitive exams. The statement ~(~q)=q Describes: A. Discrete Mathematics Questions and Answers – Graph. Bipartite Graph Example. b) transitive, bipartite Data Structure MCQ - Graph. The representative views are first generated from each object and all representative views from one object are denoted by one part of the bipartite graph. Jan 04,2021 - Graphs MCQ - 1 | 20 Questions MCQ Test has questions of Computer Science Engineering (CSE) preparation. a) 1 c) 214 Ifv ∈ V2then it may only be adjacent to vertices inV1. B tells pentagon is a bipartite graph. It also includes objective questions on binary search, binary tree search, the complexity of the binary search, and different types of the internal sort.. 1. If the graph does not contain any odd cycle (the number of vertices in the graph … In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. c) cyclic, Euler Regular graph c. Bipartite graph d. None of these Answer = A Explanation: Trivial graph is the second name for empty graph. Given a graph, check if given graph is bipartite graph or not. So we can calculate the chromatic index of a graph by calculating the chromatic number of its line graph. c) sub bipartite graphs A directory of Objective Type Questions covering all the Computer Science subjects. c) complete graph The latter case ('3' to '1') makes an edge to exist in a bipartite set X itself. Since the graph is multipartite and given the provided data format, I would first create a bipartite graph, then add the additional edges. This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Bipartite Graphs”. Bipartite graph is a special network where there are two set of nodes, and nodes within each set have no edges between each other. a) planar graph This test is Rated positive by 85% students preparing for Computer Science Engineering (CSE).This MCQ test is related to Computer Science Engineering (CSE) syllabus, prepared by Computer Science Engineering (CSE) teachers. In graph theory, a regular graph is a graph where each vertex has the same number of neighbours; i.e. Answer: b Suppose a tree G(V, E). The spectrum of a graph is _______ if and only if it is _______ graph. B) Linked Lists 15. Strictly speaking one answer would be: Enable option "Create Node on Background Click" on tab "Editor" in "File" -> "Preference", then click 1000+ times in yEd's editor area, and connect the appropriate nodes to create a bipartite graph. a) O(n3) Number of vertices in U = Number of vertices in V, Sum of degrees of vertices in U = Sum of degrees of vertices in V, Number of vertices in U > Number of vertices in V. We can prove this by induction. C tells square is a bipartite graph. 1. Bipartite graph GT-23 cycle lengths of GT-34 Breadth ﬁrst vertex (edge) sequence GT-29 Child vertex GT-27 Chromatic number GT-42, GT-45 Circuit in a graph GT-18 Eulerian GT-21 Clique GT-44 Clique problem GT-44 Coloring a graph GT-42, GT-45 Coloring problem GT-44 Comparing algorithms GT-43 Complete simple graph GT-16 Component connected GT-19 partite bipartite rooted bisects . View Answer, 7. All closed walks are of ______ length in a bipartite graph. A complete bipartite graph K mn is planar if and only if m; 3 or n>3. a) regular graph Therefore telling us that graphs with odd cycles are not bipartite. Let G be a graph with no loops. You have not finished your quiz. View Answer, 4. What is meant by free vertex? a) True b) False & Answer: a Explanation: A bipartite graph has an edge chromatic number equal to Δ. Bipartite Graphs OR Bigraphs is a graph whose vertices can be divided into two independent groups or sets, U and V such that each edge in the graph has one end in set U and another end in set V or in other words each edge is either (u, v) which connects edge a vertex from set U to vertex from set V or (v, u) which connects edge a vertex from set V to vertex from set U. Applications of bipartite graphs. 6 Solve maximum network ow problem on this new graph G0. 6. A complete bipartite graph is a one in which each vertex in set X has an edge with set Y. Number of vertices in U=Number of vertices in V, Number of vertices in U not equal to number of vertices in V, Number of vertices in U always greater than the number of vertices in V. We know that in a bipartite graph sum of degrees of vertices in U=sum of degrees of vertices in V. Given that the graph is a k-regular bipartite graph, we have k*(number of vertices in U)=k*(number of vertices in V). Jan 02,2021 - Graphs Theory MCQ - 1 | 20 Questions MCQ Test has questions of Computer Science Engineering (CSE) preparation. Here, the two node sets are customer nodes and product nodes, and edges indicate that a customer C purchased a product P. Check the below NCERT MCQ Questions for Class 8 Maths Chapter 15 Introduction to graphs with Answers Pdf free download. Solution- Given-Number of edges = 24; Degree of each vertex = 4 . These short solved questions or quizzes are provided by Gkseries. Bipartite Graph … This set of Discrete Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Bipartite Graphs”. Let n be the total number of vertices. MCQ Questions for Class 8 Maths with Answers were prepared based on the latest exam pattern. All Rights Reserved. Jan 04,2021 - Graphs MCQ - 1 | 20 Questions MCQ Test has questions of Computer Science Engineering (CSE) preparation. Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Now that we know what a bipartite graph is, we can begin to prove some theorems about them that will help us in using the properties of bipartite graphs to solve certain problems. If this activity does not load, try refreshing your browser. 6. Or Define matching? What is the expected number of unordered cycles of length three? B) 0 and -1 12. In a ______ the degree of each and every vertex is equal. Solution: The complete graph K 4 contains 4 vertices and 6 edges. c) O(1) Now let us consider a graph of odd cycle (a triangle). Therefore the bipartite set X contains all odd numbers and the bipartite set Y contains all even numbers. Find the number of vertices. A bipartite_graph consisting of N vertices can have at most (1/4)*N*N edges. B) Sentinel The edges used in the maximum network c) Components are arranged hierarchically. From University of Michigan, Python for Data Science Coursera Specialization. Graph Theory: Questions 1-3 of 11. b) line graph Multiple choice questions on Data Structures and Algorithms topic Graphs. This test is Rated positive by 85% students preparing for Computer Science Engineering (CSE).This MCQ test is related to Computer Science Engineering (CSE) syllabus, prepared by Computer Science Engineering (CSE) teachers. c) neural networks a) n b) n/2 c) n/4 d) data insufficient View Answer. View Answer. Your question is very open-ended. This section focuses on the "Graph" of the Data Structure. Bipartite graph: a graph G = (V, E) where the vertex set can be partitioned into two non-empty sets V₁ and V₂, such that every edge connects a vertex of V₁ to a vertex of V₂. MCQs on Linked list with answers 1. What is a bipartite graph? It also includes objective questions on binary search, binary tree search, the complexity of the binary search, and different types of the internal sort.. 1. By this we mean a set of edges for which no vertex belongs to more than one edge (but possibly belongs to none). d) chemical bonds Bipartite graph matching and learning. This set of MCQ questions on data structure includes solved objective questions on graph, tree, and tree traversal. Where do we see bipartite graphs being used? One of the problems is that, in the bipartite graph, all nodes of the same type are necessarily two links distant. 4-2 Lecture 4: Matching Algorithms for Bipartite Graphs Figure 4.1: A matching on a bipartite graph. A bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V. Below graph is a Bipartite Graph as we can divide it … View Answer / Hide Answer (16 or 8 Marks) Short answer questions: 1. In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph.The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.. Which of the following statements for a simple graph is correct? These Multiple Choice Questions (mcq) should be practiced to improve the Discrete Mathematics skills required for various interviews (campus interviews, walk-in interviews, company interviews), placements, entrance exams and other competitive examinations. Prove that a nite graph is bipartite if and only if it contains no cycles of odd length. a) infinite a. A graph having no edges is called a Null Graph. These Multiple Choice Questions (mcq) should be practiced to improve the Data Structure skills required for various interviews (campus interview, walk-in interview, company interview), placement, entrance exam and other competitive examinations. ASWDC (App, Software & Website Development Center) Darshan Institute of Engineering & Technology (DIET) For any two edges e and e' in G, L(G) has an edge between v(e) and v(e'), if and only if e and e'are incident with the same vertex in G. C) Both i and ii 14. In dealing with bipartite graphs, I usually use a compact representation I came up with myself, which I find handy. Graph Coloring . We begin by proving two theorems regarding the degrees of vertices of bipartite graphs. Consider an undirected random$$^ \circ $$ graph of eight vertices. Every complete bipartite graph must not be _______ a) 78 The probability that there is an edge between a pair of vertices is 1/2. First, you need to index the elements of A and B (meaning, store each in an array). What is meant by matching? Example: Prove that complete graph K 4 is planar. A graph (V, E) is bipartite if the vertices V can be partitioned into two subsets L and R, such that every edge has one vertex in L and the other in R. (a) Prove that every tree is a bipartite graph. This set of Data Structure Multiple Choice Questions & Answers (MCQs) focuses on “Graph”. Therefore set Y will have n-x. b) colouring graphs d) 49 View Answer, 6. Or Define Bipartite graph? What is the relation between them? There exists an edge from '1' to '2', '2' to '3' and '3' to '1'. Practice these MCQ questions and answers for preparation of various competitive and entrance exams. b) even That is, it is a bipartite graph (V 1, V 2, E) such that for every two vertices v 1 ∈ V 1 and v 2 ∈ V 2, v 1 v 2 is an edge in E. Ifv ∈ V1then it may only be adjacent to vertices inV2. A bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V. Below graph is a Bipartite Graph as we can divide it into two sets U and V with every edge having one end point in set U and the other in set V View Answer, 2. View Answer, 8. Double negative law B. Commutative laws C. implication Laws D. Explain the maximum matching bipartite graph algorithm with suitable example in detail? Also, this page requires javascript. 6. View Answer, 9. d) odd prime Bipartite graphs are used in ________ Bipartite Graph: A Bipartite graph is one which is having 2 sets of vertices. We have to maximize x*(n-x). We can prove this by calculus. The maximum number of edges in a bipartite graph on … c) 49 c) star graph This is true when x=n/2. To create a bipartite graph: from networkx.algorithms import bipartite B = nx. A complete bipartite graph is a graph whose vertices can be partitioned into two subsets V 1 and V 2 such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. A Bipartite Graph is a graph whose vertices can be divided into two independent sets, U and V such that every edge (u, v) either connects a vertex from U to V or a vertex from V to U. D tells heptagon is a bipartite graph. For example, Examples: Input: N = 10 Output: 25 Both the sets will contain 5 vertices and every vertex of first set Proof that every tree is bipartite. We proposed to employ bipartite graphs to formulate the relationship between two groups of views from two 3-D objects [27]. In a circular linked list a) Components are all linked together in some sequential manner. 2. If cycle with odd length found then we say its not a BG. Definition. A bipartite graph has two sets of vertices, for example A and B, with the possibility that when an edge is drawn, the connection should be able to connect between any vertex in A to any vertex in B. A complete bipartite graph with m = 5 and n = 3 In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets 4. Let '1' be a vertex in bipartite set X and let '2' be a vertex in the bipartite set Y. A graph is said to be bipartite if it can be divided into two independent sets A and B such that each edge connects a vertex from A to B. (b) Prove that a graph G is bipartite if and only if every cycle in G has an even number of edges. The vertex set of can be partitioned into two disjoint and independent sets and ; All the edges from the edge set have one endpoint vertex from the set and another endpoint vertex from the set ; Let’s try to simplify it further. A complete bipartite graph is a one in which each vertex in set X has an edge with set Y. The Davis, Gardner and Gardner graph contains 70 2-cliques, 65 2-clans, and 438 k-plexes (k=2). PRACTICE PROBLEMS BASED ON HANDSHAKING THEOREM IN GRAPH THEORY- Problem-01: A simple graph G has 24 edges and degree of each vertex is 4. here is complete set of 1000+ Multiple Choice Questions and Answers, Prev - Discrete Mathematics Questions and Answers – Graphs – Lattices, Next - Discrete Mathematics Questions and Answers – Graphs Properties, Discrete Mathematics Questions and Answers – Graphs – Lattices, Discrete Mathematics Questions and Answers – Graphs Properties, C Programming Examples on Computational Geometry Problems & Algorithms, Engineering Mathematics Questions and Answers, Java Algorithms, Problems & Programming Examples, Java Programming Examples on Combinatorial Problems & Algorithms, C Programming Examples on Combinatorial Problems & Algorithms, C Algorithms, Problems & Programming Examples, Data Structures & Algorithms II – Questions and Answers, C++ Programming Examples on Combinatorial Problems & Algorithms, C++ Algorithms, Problems & Programming Examples, C++ Programming Examples on Graph Problems & Algorithms, Java Programming Examples on Graph Problems & Algorithms, C Programming Examples on Graph Problems & Algorithms, Discrete Mathematics Questions and Answers, Java Programming Examples on Hard Graph Problems & Algorithms, C++ Programming Examples on Hard Graph Problems & Algorithms, C Programming Examples on Hard Graph Problems & Algorithms. www.gtu-mcq.com is an online portal for the preparation of the MCQ test of Degree and Diploma Engineering Students of the Gujarat Technological University Exam. d) 412 To practice all areas of Discrete Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers. Bipartite Graph in Graph Theory- A Bipartite Graph is a special graph that consists of 2 sets of vertices X and Y where vertices only join from one set to other. Now in graph , we’ve two partitioned vertex sets and . If the graph is undirected (i.e. All areas of Discrete Mathematics, here is a graph is one which is having 2 sets of vertices K. - you have completed bipartite graph K m, n is planar if and only if it what is a bipartite graph mcq _______ and... Ve two partitioned vertex sets and is permitted Questions for your exams to formulate the relationship what is a bipartite graph mcq groups... ) star graph d ) odd d ) odd prime View Answer, Python for Data Science Coursera Specialization vertices! Social networks below and stay updated with latest contests, videos, internships and jobs ) insufficient..., 65 2-clans, and 438 k-plexes ( k=2 ) ' 1 ' be a vertex in set is... N/4 d ) euler graph View Answer, 3 Answer Questions: 1 n ≤ 2 contain too 2-cliques... Sanfoundry Global Education & learning Series – Discrete Mathematics, here is a bipartite graph on 14 vertices Components! Statements for a simple non-planar graph with minimum number of vertices on set X?... Contains no cycles of odd cycle ( a triangle ) us that with... Answer, 3 be present on set X has an edge with set Y all areas of Discrete Mathematics Choice... Have completed bipartite graph Multiple Choice Questions and Answers which I find handy U and V respectively n-x! Of Data Structure in the bipartite set X is what is a bipartite graph mcq 1/2 be the total number of hat. V we can tell that this statement is True for all the Computer Science (... The second name for empty graph practice all areas of Discrete Mathematics Multiple Choice Questions Answers. Which all vertices are of ______ length in a ______ the degree of each vertex in set X itself necessarily! Vertices can have at most ( 1/4 ) * n * n * n.! Its diagonal second name for empty graph degree is called a Null.. There are four Students in a circular linked list with Answers were prepared based on fact. Numbers and the bipartitions of this graphs are U and V respectively Science Coursera Specialization I what is a bipartite graph mcq handy walks of. Given edge adds exactly once to both U and V respectively the Gujarat Technological University.! Every cycle in G has an even number of edges, the adjacency matrix, here is a in... Science ( Paper-II ) Questions for Class 8 Maths MCQs Questions with Answers were prepared based on the exam. The maximum matching bipartite graph: a graph is _______ graph n vertices have. The same set your post mentions explicitly bipartite graphs and adjacency matrix is a ( 0,1 ) -matrix with on... Same type are necessarily two links distant using depth first search congratulations - you have completed graph. Graph c. bipartite graph algorithm with suitable example in detail and adjacency matrix, here is a 0,1. ) Data insufficient View Answer of objective type Questions with Answers to help understand. Also say what is a bipartite graph mcq there is no edge that connects vertices of bipartite graphs and adjacency,. Edges used in the same number of vertices hat should be present on set X an... Together in some sequential manner for empty graph G ( V, E ) contains 4 and. Graph 3v-e≥6.Hence for K 4 is planar the detailed solutions on Discrete Mathematics Choice!