Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from v0 to v1 , then there is a simple path in G from v0 to v1

Q: Let G = (V, E) be a planar graph such that |V |< 19. Show that if the length of the shortest cycle…

A: For a given graph G=V,E where V≤19 that shortest cycle in G is 5, then we need to prove that G has a…

Q: Let G be a DAG with exactly one source r and such that for any vertex v there exists a unique…

A: Given G is a directed acyclic graph (DAG).It is also given that G has exactly one source r and for…

Q: Suppose that an edge e ::= {v1, v2 } is on a cycle in a graph G where v1, v2 are vertices of G. If e…

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Q: Prove that if G is a graph with no isolated vertices, then α'(G) +β'(G) = n(G)

A: Here α'(G) is largest size of matching in G. β'(G) is the smallest size of an edge cover. We have to…

Q: Show that For n > 1 let G, be the simple graph with vertex set V(Gn) = {1,2,., n} in which two…

A: A simple connected graph G is said to be planar if we can draw the graph in the plane with no two…

Q: 4. Let n eN with n > 1, V = Z and set E = {{r, y} : x, y E Z and |r – y| = n}. Define G, = (V, E).…

A: Given :                  n∈ℕ and n≥1                V = ℕ ( set of integers)and the set, E = x,y :…

Q: Give an example to show that if P is a (u, v)-path in a 2-connected graph G, then G does  not…

A: To show that if P is a u,v-path in a 2-connected graph G, then G does not necessarily contain a…

Q: Let G be a connected graph that has exactly 4 vertices of odd degree: V1, V2, V3 and V4. Show that…

A: Let G be a connected graph which has 4 vertices of odd degree. The objective is to show that there…

Q: Let G be a connected graph with at least one edge and F ⊆ E(G) be an edge cut. Prove that F is a…

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Q: Determine whether the following statement is true or false. A Hamilton path must contain every…

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Q: Prove that connecting two nodes u and v in a graph G by a new edge creates a new cycle if and only…

A: Given, Connecting two nodes u and v in a graph G by a new edge creates a new cycle such that if and…

Q: Draw a graph G with vertex and edge set  , where the endpoints of e1 are v1 and v2, the endpoints of…

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Q: Let n € N with n > 2 and let G₁ = (V, E) be the graph with vertex set V = {₁, V2, ..., V2n} and edge…

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Q: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

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Q: Prove that every graph G on {1, ... n} which is acyclic and has n – 2 edges has exactly 2 connected…

A: A graph with v vertices and e edges has at least v-e connected components. By induction on e, If e=0…

Q: simple path in G from v0 to v1

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Q: Give an example to show that if P is a (u, v)-path in a 2-connected graph G, then G does not…

A:

Q: Let G be a connected graph that has an Euler tour. Prove or disprove the following statements. If G…

A: Euler Circuit: A closed walk in a graph G that uses every edge exactly once is called an Euler…

Q: Let G = (V, E) be a finite bipartite graph with bipartition (A, B) where DI

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Q: Let D be an n-vertex digraph with no cycles. Prove that V (D) can be ordered as v1, . . . , vn such…

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Q: 5. If G is a simple graph with d(v) 2 k, Vv € V(G), then G contains a path of length at least k. If…

A: For k = 0 the statement is trivial because for any v ∈ V the sequence v (of one term in V ) forms a…

Q: (b) Show that G may not necessarily have three cycle subgraphs by giving an example of a graph G,…

A: Consider the equation

Q: Let G be a DAG with exactly one source r and such that for any vertex v there exists a unique…

A: Given: G is a DAG(direct Acylic Graph) with exactly once source r. There exist a unique path from r…

Q: Let G be a graph with vertex set V(G) = {v1, v2, V3, V4, V5, V6, V7} and edge set E(G) = {v1v2,…

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Q: Given a graph G = (V, E) such that V = {1, 2, 3, 4, 5, 6} and |E| = 10. Prove that there exists a…

A: Given - A Graph G = (V, E) such that V = { 1, 2, 3, 4, 5, 6} and E = 10 . To prove - There exists a…

Q: The Euler path of the graph K₁ is h O a O b O c Od g C e К1 d (h, a, b, f, g, e, d, j, c, a) (g. f,…

A: The only way to identify Euler path ~ Use every edge EXACTLY once + start &amp; end at different…

Q: 4. Prove: a) In a graph, if two vertices are connected by a walk then they are connected by a path.…

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Q: c) Prove that if a plane graph G has n vertices, e edges, f faces and k components then n- e+f =k+1.

A: We can prove Euler's formula (n-e+ f = k+1) works generally for anyconnected simple planar graph.so,

Q: Suppose G has V vertices and E edges. Let M and m denote, respectively, the maximum and minimum of…

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Q: Let's consider an undirected graph G = that for any vertices, i, j E I and there is no edge between…

A: Given: G = V , E is an undirected graph. An independent subset is a subset I ⊂ V such that for any…

Q: Consider the graph G in Fig. 8-58. Find: (a) degree of each vertex (and verify Theorem 8.1); (b) all…

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Q: Prove that every simple graph G= (V, E) with |V] > 2 has at least two vertices of equal degree. %3D…

A: We have a simple graph, with V≥2, that is, the number of vertices is greater than or equal to 2.…

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

A:

Q: Let H be a connected planar graph with at least 3 vertices. Prove that f is greater than or equal to…

A:

Q: Prove that every graph G on {1, ...n} which is acyclic and has n – 2 edges has exactly 2 connected…

A: A graph is said to be acyclic if it has no cycles and acyclic if it contains no cycle. For a graph…

Q: Prove : for r belongs to Z+, every r connected graph on an even number of vertices with no induced…

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Q: Prove that if  v0 and v1  are distinct vertices of a graph  G  = (V,E) and a path exists in G from…

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Q: For each of the two directed graphs below,

A: Given, Diagram of two directed graphs and . We have to determined that the given two directed…

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

A: Solution :-

Q: Which of the following graphs does NOT contain an Euler path? E В. E С. D.

A: An Euler path in a graph is a path that uses every edge exactly once and this path starts and ends…

Q: Let G=(V, E) be a graph such that • V| ≥ 3, G has exactly 2 leaves, and in G all the non-leaf…

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Q: Let G be a n vertices graph ( n 2 3 ), such that d(u) + d(v) 2 n, for every pair of non adjacent…

A: Let us start proving step by step

Q: Let n e N with n > 2 and let G, = (V, E) be the graph with vertex set V = {v1, v2, . . . , v2n} and…

A: Since you have posted a question with multiple subparts, we will solve the first three subparts…

Q: Prove that if G is a graph of order p and has size greater than ¹) then G is connected. 2

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Q: For any connected graph G of order 54, if x is a vertex of G, then the number of components of G-x…

A: Given: To explain the given statement as follows,

Q: 7. Give an (i) a non-planar graph that is not homeomorphic to K5or K3,3 example of

A: To find: A graph which is non planar and not homomorphic to K5 or K3,3. If the edges of the graph…

Q: Let G = (V, E) be a simple undirected graph without loops, where |V| > 2. Let 8(v) denote the degree…

A: a) The statement is there does not exist u and v such that δu=0 and δv=V-1. First let us consider a…