What is wrong with the following "proof"? In addition to finding a counterexample, you should explain what is fundamentally wrong with this approach, and why it demonstrates the danger of build-up error. False Claim: If every vertex in an undirected graph has degree at least 1, then the graph is con- nected. Proof? We use induction on the number of vertices n ≥ 1. Base case: There is only one graph with a single vertex and it has degree 0. Therefore, the base case is vacuously true, since the if-part is false. Inductive hypothesis: Assume the claim is true for some n ≥ 1. Inductive step: We prove the claim is also true for n + 1. Consider an undirected graph on n vertices in which every vertex has degree at least 1. By the inductive hypothesis, this graph is connected. Now add one more vertex x to obtain a graph on (n+1) vertices, as shown below. n-vertex graph All that remains is to check that there is a path from x to every other vertex z. Since x has degree at least 1, there is an edge from x to some other vertex; call it y. Thus, we can obtain a path from x to z by adjoining the edge {x,y} to the path from y to z. This proves the claim for n + 1.

Computer science question help

Full explain this question very fast solution sent me step by step  Not :- Text typing work only do u and all part work  Not allow paper work  Thanks

There are many applications of Shortest Path Algorithm. Consider the problem of solving a jumbled Rubik's Cube in the fewest number of moves. I claim that this problem can be solved using a Shortest Path Algorithm. Determine whether this statement is TRUE or FALSE. NOTE: if you want to check if this statement is TRUE, think about how the Rubik's Cube Problem can be represented as a graph. What are the vertices? Which pairs of vertices are connected with edges? What is your source vertex and what is your destination vertex? How would Dijkstra's Algorithm enable you to find the optimal sequence of moves to solve a jumbled cube in the fewest number of moves?

Is this statement about the chromatic number true? Please give some prove and brief explaination: If A and B are graphs on the same vertex-set, then χ(A ∪ B) ⩽ χ(A) + χ(B).

Could you also do question one please

Question 1: In graph theory, a graph X is a "complement" of a graph F if which of the following is true?   Select one:   a. If X is isomorph to F, then X is a complement of F.   b. If X has half of the vertices of F (or if F has half of the vertices of X) then X is a complement of F.   c. If X has the same vertex set as F, and as its edges ONLY all possible edges NOT contained in F, then X is a complement of F.   d. If X is NOT isomorph to F, then X is a complement of F.   Question 2: Which statement is NOT true about Merge Sort Algorithm:   Select one:   a. Merge Sort time complexity for worst case scenarios is: O(n log n)   b. Merge Sort is a quadratic sorting algorithm   c. Merge Sort key disadvantage is space overhead as compared to Bubble Sort, Selection Sort and Insertion Sort.   d. Merge Sort adopts recursive approach