Prove: Given a graph G = (V,E) of degree 3 and free of circles and swords of length 1, if G contains a sword (h,v1, v2,, vl) where l 4 the simple branching operation on h contributes no more than 20.310 to the total time complexity.
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Prove: Given a graph G = (V,E) of degree 3 and free of circles and swords of length 1, if G contains a sword (h,v1, v2,, vl) where l 4 the simple branching operation on h contributes no more than 20.310 to the total time complexity.
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- Prove:Given a graph G = (V,E) of degree bounded by 3 and free of circles and swords of length 1, if G contains a sword (h,v1, v2, ··· , vl) where l ≥ 4, the simple branching operation on h contributes a factor of at most 20.310 to the overall time complexity. Prove the following.(Note: Provide each an illustration for verification of results)Let H be a spanning subgraph of a graph G.i. If H is Eulerian, then G is Eulerian.ii. If H is Hamiltonian, then G is HamiltonianGive, with justification, a formula for the minimum number of edges that must be added to a general connected graph to make it have an Euler tour, provided we allow multiple edges (that is, two vertices can be joined by more than one edge).
- For the following simple graphs G=(V,E) (described by their vertex and edge sets) decide whether they are bipartite or not. If G is bipartite, then give its bipartition, and if it is not explain why. 1) V={a,b,c,d,e,f,g}, E={ag,af,ae,bf,be,bc,dc,dg,df,de} 2) V={a,b,c,d,e,f,g,h}, E={ac,ad,ah,bc,bh,bd,ec,ef,eg,hf,hg}Say that a graph G has a path of length three if there exist distinct vertices u, v, w, t with edges (u, v), (v, w), (w, t). Show that a graph G with 99 vertices and no path of length three has at most 99 edges.Prove that in each component of the context of any graph, the number of vertices of an odd degree is always even.
- I really want to learn. Please explain too. For each graph representation, what is the appropriate worst-case time complexity for checking if two distinct vertices are connected. The choices are: O(1), O(V), O(E), or O(V+E) Adjancy Matrix = ____ Edge List = ____ Adjacency List = ____For each graph representation, select the appropriate worst-case complexity: Adjacency Matrix: ________ Edge List: ________ Adjacency List: _________ Choices: O(V+E), O(E^2), O(V^2), O(E) Explain why.not handwritten Find with proof every graph G for which ex(n, G) is defined for all positive integers n andthere exists a constant c such that ex(n, G) = ex(m, G) for all integers m, n > c.
- A given graph G of order 5 contains at least two vertices of degree 4. Assume that notall vertices in G are even. Find all possible degree sequences of G, in non-increasingorder; and for each case, construct all such G which are not isomorphic.If a graph G = (V, E), |V | > 1 has N strongly connected components, and an edge E(u, v) is removed, what are the upper and lower bounds on the number of strongly connected components in the resulting graph? Give an example of each boundary case.For each graph representation, select the appropriate worst-case complexity: Adjacency Matrix: ________ Edge List: ________ Adjacency List: _________ Choices: O(V+E), O(E^2), O(V^2), O(E)