How does DeMorgans law show that the negation of a k-map will give the same answer as the original kmap?

Algorithm Element as a Strong Generator ProductInput: a group G operating on f = 1,2,.....n; an element g of G; a base and strong generating set for G; Schreier vectors v I 1 I k, for the stabiliser chainOutput: a symbolic product, a word that expresses g in terms of the strong generators.

The problem Monochomatic-Subgraph-Avoidance takes as input two undirected graphs Fand G. It asks where F can be colored with two colors (say, red and blue) that does not containa monochomatic (all-red or all-blue) G as a subgraph. (Note that when G has two nodes and oneedge between them, this is equivalent to the 2-coloring problem of asking whether F can be coloredso that no neighbors have the same color.)Next, show that Monochomatic-Subgraph-Avoidance is contained in one of the classes inthe second level of the polynomial hierarchy.  ALso provide an expression using qualifiers. YOU ALSO NEED TO PROVIDE A CLEAR EXPRESSION USING THE QUALIFIERS.PLEASE GIVE CLEAR EXPLAINATION.

Write a functionIsItASpanningSubgraph (G,H) which takes as input two graphs in nx format (not as dictionaries) and outputs True if H is a spanning subgraph of G and False otherwise.

Make java program that gives results to determine if a connected simple graph has independent sets & identify the minimum set & maximum independent set. Output of the program: - independent sets - maximum independent set - minimum independent set

Exercise c. Strict matchingConsider the following generalization of the maximum matching problem, which we callStrict-Matching. Recall that a matching in an undirected graph G = (V, E) is a setof edges such that no distinct pair of edges {a, b} and {c, d} have endpoints that areequal: {a, b} ∩ {c, d} = ∅. Say that a strict matching is matching with the propertythat no pair of distinct edges have endpoints that are connected by an edge: {a, c} ̸∈ E,{a, d} ̸∈ E, {b, c} ̸∈ E, and {b, d} ̸∈ E. (Since a strict matching is also a matching, wealso require {a, b} ∩ {c, d} = ∅.) The problem Strict-Matching is then given a graphG and an integer k, does G contain a strict matching with at least k edges.Prove that Strict-Matching is NP-complete.

Hey, Kruskal's algorithm can return different spanning trees for the  input Graph G.Show that for every minimal spanning tree T of G, there is an execution of the algorithm that gives T as a result.   How can i do that? Thank you in advance!