Let G be a planar graph in which every face is either a triangle or a pentagon. If G has 12 pentagons and 20 triangles, how many vertices does G have?

Q: Consider the graph as shown in figure and determine the following: Pendant vertices, pendant edges,…

A: 1. Pendant vertices are d,e,f,g Pendant edges are ad edge,eb edge,fb edge,gb edge, Odd vertices are…

Q: Let G be a simple undirected graph (without multiple edges and loops) with  n vertices and m edges.…

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Q: Let A and B each be sets ofN labeled as vertices, and consider the bipartite between A and B 1. What…

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Q: Which of these graphs are trees? b) d)

A: A tree is a connected graph which has no cycles in it. Equivalently, A connected  graph of n…

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

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Q: 1. Suppose G is a planar graph with four faces of lengths 3, 4, 5, 6. How many edges are there in G?

A: Answer :-  a). Suppose G is a planar graph with faces of length 3, 4, 5, 6. How many edges are there…

Q: Let G be a (p, q) graph. Prove that the sum of the degrees of the vertices of G equals twice the…

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Q: Suppose that a graph G has n vertices, and all vertices have degree greater than or equal to“,.…

A: Let G be a graph with n vertices. Given that all vertices have degree greater than or equal to n-12.…

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

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Q: 13. Determine whether the following graph is strongly connected. b.

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Q: Let G be a simple graph with n vertices, which is regular of degree d. By considering the number of…

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Q: Show that if a simple graph G has n vertices and at least n edges, then G must contain a cycle.

A: We have to Show that if a simple graph G has n vertices and at least n edges, then G must contain a…

Q: Show that if u and v are the only odd-degree vertices in G, then there is a uv path G.

A: We prove this by induction on the number of vertices in G.   Case 1:   If the graph consists of…

Q: Show, by drawing, that the following graphs are planar: the wheel W5;

A: The graph of W5 is in next step

Q: Give an example of a graph with an infinite number of vertices that has a finite chromatic number.…

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Q: 1.1 How many simple graphs are there of four vertices? Draw all of them.

A: As per our guidlines we are supposed to solve only one question that is the first problem. For the…

Q: Show that if a simple graph G ha

A: To prove that if a simple graph G has n vertices and at least n edges, then G must contain a cycle.

Q: In each part, draw a (simple) graph G with the given properties or argue that no such graph exists.…

A: We have to find a graph having 6 vertices and exactly 2 biconnected components.

Q: If G is a connected planar graph where e = 3v - 6, show that every region is triangular (has three…

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Q: If a graph G has n vertices, all of which but one have odd degree, how many vertices of odd degree…

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Q: For which values of n does the graph Cn have an Euler circuit?

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Q: 1. Which of these graphs are trees? а) b) (p f) 2. Which of these graphs are trees? а) b) c)

A: There are two questions given . But according to rules of Bartleby, only the first question can be…

Q: The ff. graph is planar:

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Q: O Let G be a such that graph with 10 verti ces 8CG) = ACG) = 2 How many edges does G have?

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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: In each part, draw a (simple) graph G with the given properties or argue that no such graph exists.…

A: As per the question we have to draw a connected graph having 6 vertices and 5 articulation points.

Q: Let G be a graph with 10 vertice and 5 components. If a vertex is removed from G, the number of…

A: Answer: 9 and 4

Q: Let G = (V, E) be a k-connected graph, let æ E V, and let B C V\{x} such that |B| > k. Show that…

A: Fan: Let v∈V(G) and let A∈V(G)\{v}. A v,A-fan of size k is a collection of k paths P1,P2,...,Pk so…

Q: The number of edges in a graph with 8 vertices each of degree 5 is 10 O 20 O 7 O 40

A: We have to determine the number of edges in the graph with 8 vertices of degreee 5 .

Q: 17. List all the vertices that are adjacent to vertex A.

A: This question is related to graph and figures, we will solve it using given information.

Q: Given an acyclic graph G = (V, E) return the longest path from two connected vertices: u, v. Prove…

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Q: Pick a set of five vertices of G and draw the subgraph induced by that set of vertices G:

A: Let GV,E be a graph with vertex set V and edge set E.  A vertex-induced subgraph HV1,E1 is a…

Q: Let G be a planar graph. Use Euler's Formula to prove that G must have a vertex of degree 5 or less.…

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Q: 5. For any graph G with six vertices, then G or its complement Ġ contains a triangle.

A: The solution for the above question is as shown below.

Q: Let G be a graph with vertex set VG and edge set EG. Let V C VG and EC EG. Is there always a…

A: Consider the given information.

Q: prove If G is a plane graph with n vertices, m edges and r regions, then n - m + r = 1 + k(G).

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

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Q: Let G be a graph with exactly 10 vertices and 27 edges. Suppose that each vertex has degree 3, 5, or…

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Q: Find the sum of the degrees of the vertices of each graph in Exercises 1-3 and verify that it equals…

A: We know that, Degree of vertex is the number of edges passing through that vertex. We need to…

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: Show that any graph with at most 8 vertices is planar?

A: Given: planar graph

Q: For each undirected graph in Exercises 3–9 that is notsimple, find a set of edges to remove to make…

A: Given That : Unidirectional graphs are given that may or may not be simple  To Find : A set of edges…

Q: Let G be a simple graph with 20 vertices. Suppose that G h ponents, and every pair of distinct…

A: Given: Simple graph G with 20 vertices. G has two-component with distinct vertices that satisfy…

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 if a plane graph G has n vertices, e edges, f faces and k components, then n−e+ f = k +1.

A: Euler’s Theorem: According to Euler's Theorem, V-E+F=2 Where, V is the number of vertices. E is the…

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…