Suppose a planar graph G is not connected and has exactly 2 components. What is the value of n – m+ f now where n, m, f is the number of vertices, edges, and faces of a planar drawing of G respectively?

Q: Prove that there must be at least two vertices with the same degree in a simple graph

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Q: 2. Show that any graph with at most 8 vertices is planar.

A: Topic:- planer graph

Q: 3. а) Define the following terms as used in graph theory: i) regular graph ii) simple circuit

A: A graph is called a regular graph as graph it degree of each vertex. is equal. A graph is called k…

Q: Let G be a simple connected planaı 13 vertices and 19 edges. Then, the number of faces in the planar…

A: Given that, G is a simple connected planar graph with 13 vertices and 19 edges.

Q: Let G be a graph such that every vertex has degree 4 and the number of edges is 12. How many…

A: Simply keep in mind that vertex degree multiplied by number of vertices is two times number of…

Q: Suppose a graph G is regular of degree r, where r is odd. (a) prove that G has an even even number…

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Q: Suppose G is a simple graph (no loops; no parallel edges) with n = 14 vertices and m = 7 edges. What…

A: GIVEN  vertices n=14 edges m=7 EXPLANATION as we know  every graph has atleast n-m component…

Q: Can we draw a planar graph with n= 7 and e = 12 such that each region is bounded by exactly 3-edges…

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Q: Show that every graph with n vertices and k edges, n> k has n – k components.

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Q: Give an example of a graph with five vertices of degree 1, and six vertices of degree 2. If you…

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Q: 1.Show that K3,3 is non planner graph

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Q: In Problem ,draw a complete graph so that it has the type of symmetry indicated.

A: Given that, This is the graph is an upper horizontal half parabola whose vertex at -4, 0 crosse…

Q: Give an example of a graph G of order 6 and size 10 such that the minimum and maximum degree of a…

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Q: Suppose a (simple) graph G = (V, E) is regular with odd vertex degree. Prove that the number of…

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Q: Consider the following directed graph. a. Give the indegree of each vertex. b. Give the outdegree of…

A: The vertices are {a , b , c , d , e} (a) The indegree of the vertex is the number of edges coming to…

Q: Suppose that we have a graph with at least two vertices. Show that it is not possible that all…

A: Suppose, we have a graph with at least two vertices.

Q: Let G be a planar graph in which every face is either a triangle or a pentagon. If G has 12…

A: It is given that Let G be a planar graph in which every face is either a triangle or a pentagon. If…

Q: Draw a bipartite graph with a vertex partition into the two sets {v1, v2, v3, v4} and {v5, V6, v7}…

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Q: Show that if G is a simple graph with n vertices (where n is a positive integer) and each vertex has…

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Construct a graph with vertices R,S,T,U whose degrees are 0,0,1,1. What is the edge set?

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Q: Show that every simple finite graph has two vertices of the same degree.

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Q: You have a simple graph G with three vertices and three edges. How many subgraphs can be made from…

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Q: Find a subgraph H of the graph G given below that has 4 vertices and 3 edges.

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Q: (a) G is a connected graph with 6 vertices and 5 articulation points.

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Q: Show that any connected simple planar graph on n vertices must have at most n/2 vertices of degree…

A: Consider the equation

Q: 2.29. Is this graph planar (the edges cross each other in the middle but there is no vertex there)?

A: Planar graph: When a connected graph can be drawn without any edges crossing it is called planar.…

Q: 3 5

A: Isomorphism of graph: Let G and H be two graphs. They are said to be isomorphic if there exists a…

Q: If G is a graph, then the sum of its vertex degrees is nonnegative and even.

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Q: 2. Suppose G is a planar graph having 67 edges. If each face of G has length at least 6, then what…

A: From the given information. The number of edges (e) = 67 The number of vertices at each face (v) = 6

Q: 9. Verify that Euler's formula does not hold for a graph that is not connected: count the vertices,…

A: Euler's formula for v vertices, e edges and f faces is: v-e+f=2 For the given graph we have vertices…

Q: If G is a simple graph with at least two vertices. prove that G must contain two or more vertices of…

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Q: show that every simple finite graph has two vertices of the same degree

A: Solution: Suppose G is a finite simple graph with n vertices.  Since G is simple, no vertex is…

Q: Show that every simple planar graph with n vertices, where n ≥ 3, is a subgraph generating a…

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Q: Prove the following. 1. Prove that the number of odd vertices of a graph is even.

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Q: The number of non-isomorphic simple graphs on 8 vertices, all of which have degree 2, is Select one:…

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Q: True or False: There exists a simple graph with 9 vertices each of degree 5. O True O False

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Q: Determine if the following pair of graphs are isomorphic. If so, give the vertex pairings; if not,…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: If G is a (not necessarily simple) graph with n vertices where each vertex has degree greater than…

A: Definitions: Simple graph: A simple graph is an undirected graph with no parallel edges and…

Q: 12. If G is a simple graph with vertices a, b, c, d, e and d(a) = d(b) = d(c) = 3 (d = degree), then…

A: Here we apply Hand-shaking lemma for simple graph.

Q: 2.30. Explain why every subgraph of a planar graph must also be planar.

A: Planar graph: When a connected graph can be drawn without any edges crossing it is called planar.…

Q: Consider a graph G where the vertices are points (a, b) with integer coordinates, and {(ai, bi),…

A: Given: Let us consider graph G, in which the vertices are integer-coordinated points (a,b), and (ai,…

Q: Prove that in any simple graph G with n vertices and m edges, 2m ≤ n^2 − n.

A: We need to prove that in any simple graph G with n vertices and m edges, 2m≤n2−n

Q: Suppose G is a simple undirected graph with n vertices. If G is self-complementary, prove that…

A: ↦A Simple undirected graph G is meant by a graph which have no multiple loop. there exist at most…

Q: Is it possible for two different (non-isomorphic) graphs to have the same number of vertices and the…

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Q: Give an example of a non-planar graph that is not homeomorphic to K5or K3,3.

A: We can solve this using definition of planar graph.

Q: The complementary graph G' of a simple graph G hasthe same vertices as G. Two vertices are adjacent…

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Q: Let G be a simple connected plane graph with 6 vertices. (a) (b) (c) (d) What is the largest number…

A: Note - Acc. to Bartleby guidelines, we are supposed to do only first 3 subparts . If you want more…

Q: Let G be a simple undirected graph with 16 edges. If all vertices have degree at most four, which of…

A: Let G be a simple graph with n vertices and e edges. Use the following formula, to determine the…

Q: prove that the degree of a regular complete tripartite graph kr,s,t with n vertices is given by 2n/3

A: Since the graph is tripartite with vertex division of r, s, t, we have three clusters of vertices…

Q: Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph…

A: Given that, v and w are non-adjacent vertices in G. (1): G+e the simple graph formed by adding the…