Prove 1 For a graph G = (V, E), a forest F is any set of edges of G that does not contain any cycles. M = (E, F) where F = {F ⊆ E : F is a forest of G} is a matroid.
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Prove 1 For a graph G = (V, E), a forest F is any set of edges of G that does
not contain any cycles. M = (E, F) where F = {F ⊆ E : F is a forest of G} is a
matroid.
Step by step
Solved in 2 steps
- Let G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is incident to at least one vertex in C. We say that a set of vertices I form an independent set if no edge in G connects two vertices from I. For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of the 20 edges in the graph has at least one endpoint in C, and I = = [a, c, i, k] is an independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik. 2a In the example above, notice that each vertex belongs to the vertex cover C or the independent set I. Do you think that this is a coincidence? 2b In the above graph, clearly explain why the maximum size of an independent set is 5. In other words, carefully explain why there does not exist an independent set with 6 or more vertices.1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) If G contains a vertex of degree k, show that |L| > k. (b) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (c) Prove that either there is a vertex v e V(G) such that f(v) = v or there is an edge {x, y} € E(G) such that {f(x), f(y)} = {x, y}. (Hint: Induction on |V(G)|; in the induction step consider a restriction of ƒ to a subset of vertices.)Be G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is the greatest distance between two of its vertices. a) Show that if A(G) 24 then A(G) <2. b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no neighbors.
- Let G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-setThe minimum vertex cover problem is stated as follows: Given an undirected graph G = (V, E) with N vertices and M edges. Find a minimal size subset of vertices X from V such that every edge (u, v) in E is incident on at least one vertex in X. In other words you want to find a minimal subset of vertices that together touch all the edges. For example, the set of vertices X = {a,c} constitutes a minimum vertex cover for the following graph: a---b---c---g d e Formulate the minimum vertex cover problem as a Genetic Algorithm or another form of evolutionary optimization. You may use binary representation, OR any repre- sentation that you think is more appropriate. you should specify: • A fitness function. Give 3 examples of individuals and their fitness values if you are solving the above example. • A set of mutation and/or crossover and/or repair operators. Intelligent operators that are suitable for this particular domain will earn more credit. • A termination criterion for the…Show that we can use a depth-first search of an undirected graph G to identify theconnected components of G, and that the depth-first forest contains as many treesas G has connected components. More precisely, show how to modify depth-firstsearch so that it assigns to each vertex an integer label :cc between 1 and k,where k is the number of connected components of G, such that u:cc D :cc ifand only if u and are in the same connected component.
- An undirected graph G = (V, E) is called “k-colorable” if there exists a way to color the nodes withk colors such that no pair of adjacent nodes are assigned the same color. I.e. G is k-colorable iff there existsa k-coloring χ : V → {1, . . . , k}, such that for all (u, v) ∈ E, χ(u) ̸= χ(v) (the function χ is called a properk-coloring). The “k-colorable problem” is the problem of determining whether an input graph G = (V, E) isk-colorable. Prove that the 3-colorable problem ≤P the 4-colorable problem.Consider the graph G with n vertices. We define a Covid-Friends set as a set of vertices none of which is connected to another vertex in the set (they are social distancing!). In other words, any subgraph of G with Covid-Friends vertices has no edge. Clearly, an empty set or a set with a single vertex is a Covid-Friend vertex. However, given a graph, we want to find a such set with maximum number of vertices. (a) works - it may not even give the optimal solution!). Justify why your algorithm makes sense (in terms of the goal: maximizing the Covid-Friend set solution). Design (explain in words) a greedy algorithm for this problem (no need to prove itRecall the Clique problem: given a graph G and a value k, check whether G has a set S of k vertices that's a clique. A clique is a subset of vertices S such that for all u, v € S, uv is an edge of G. The goal of this problem is to establish the NP-hardness of Clique by reducing VertexCover, which is itself an NP-hard problem, to Clique. Recall that a vertex cover is a set of vertices S such that every edge uv has at least one endpoint (u or v) in S, and the VertexCover problem is given a graph H and a value 1, check whether H has a vertex cover of size at most 1. Note that all these problems are already phrased as decision problems, and you only need to show the NP-Hardness of Clique. In other words, we will only solve the reduction part in this problem, and you DO NOT need to show that Clique is in NP. Q4.1 Let S be a subset of vertices in G, and let C be the complement graph of G (where uv is an edge in C if and only if uv is not an edge in G). Prove that for any subset of vertices…
- 5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…How do I do this? We say a graph G = (V, E) has a k-coloring for some positive integer k if we can assign k different colors to vertices of G such that for every edge (v, w) ∈ E, the color of v is different to the color w. More formally, G = (V, E) has a k-coloring if there is a function f : V → {1, 2, . . . , k} such that for every (v, w) ∈ E, f(v) 6= f(w).3-Color problem is defined as follows: Given a graph G = (V, E), does it have a 3-coloring?4-Color problem is defined as follows: Given a graph G = (V, E), does it have a 4-coloring?Prove that 3-Color ≤P 4-Color.(hint: add vertex to 3-Color problem instance.)Square graph. The square of a directed graph G = (V,E) is the graph G^2 =(V,E^2)suchthat(u,w)∈E^2 if and only iff or some v∈V both(u,v)∈E and (v, w) ∈ E; that is, there is a path of exactly two edges. (i) Suppose that G is represented by an adjacency list. Describe an efficient algorithm for computing G^2 from G. (ii) Suppose that G is represented by an adjacency matrix. Describe an efficient algorithm for computing G^2 from G.