Question 3. Draw a graph G = (V. E) on 10 nodes (vertices) with degrees 1.1.1.1.1.1.1.1.5,...
Problem 1: Given a graph G (V,E) a subset U S V of nodes is called a node cover if each edge in E is adjacent to at least one node in U. Given a graph, we do not know how to find the minimum node cover in an efficient manner. But if we restriet G to be a tree, then it is possible. Give a greedy algorithm that finds the minimum node cover for a tree. Analyze its correctness...
3. Given graph G = (V,E), prove that the following statements are equivalent. [Note: the following statements are equivalent definitions of a "tree graph".] 1) There exist exactly one path between any of two vertices u, v EV in the graph G 2) Graph G is connected and does not contain any cycles. 3) Graph G does not contain any cycles, and a cycle is formed if any edge (u, v) E E is added to G 3. Given graph...
Let G be a tree with v vertices which has precisely four vertices of degree 1 and precisely two vertices of degree 3. What are the degrees of the remaining vertices? Let G be a tree with v vertices which has precisely four vertices of degree 1 and precisely two vertices of degree 3. What are the degrees of the remaining vertices?
Recall the definition of the degree of a vertex in a graph. a) Suppose a graph has 7 vertices, each of degree 2 or 3. Is the graph necessarily connected ? b) Now the graph has 7 vertices, each degree 3 or 4. Is it necessarily connected? My professor gave an example in class. He said triangle and a square are graph which are not connected yet each vertex has degree 2. (Paul Zeitz, The Art and Craft of Problem...
3. Given graph G-(V, E), prove that the following statements are equivalent. [Note: the following statements are equivalent definitions of a "tree graph".] 4) Graph G is connected, but would become disconnected if any edge (u,v) E E is removed from G 5) Graph G is connected and has IV 1 edges 6) Graph G has no cycles and has |V| -1 edges.
Question 16. A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. (a) Draw a maximal plane graphs on six vertices. (b) Show that a maximal plane graph on n points has 3n − 6 edges and 2n − 4 faces. (c) A triangulation of an n-gon is a plane graph whose infinite face boundary is a...
Discrete Math Create a graph with 4 vertices of degrees 2, 2, 3, 3 or explain why no such graph exists. If the graph exists, draw the graph, label the vertices and edges. To answer the question in the box below, write the vertex set, the edge set, and the edge-endpoint function as shown on page 627 of the text. You can copy (Ctrl-C) and paste(Ctrl-V) the table to use in your answer if you like. Vertex set- Edge set...
A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. A maximal plane graph is a plane graph G = (V, E) with n-3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. (a) Draw a maximal plane graphs on six vertices b) Show that a maximal plane graph...
File Edit Format View Help Graphs and trees 4. [6 marks] Using the following graph representation (G(V,E,w)): v a,b,c,d,e,f E fa,b), (a,f),fa,d), (b,e), (b,d), (c,f),(c,d),(d,e),d,f)) W(a,b) 4,W(a,f) 9,W(a,d) 10 W(b,e) 12,W(b,d) 7,W(c,d) 3 a) Draw the graph including weights. b) Given the following algorithm for Inding a minimum spanning tree for a graph: Given a graph (G(V,E)) create a new graph (F) with nodes (V) and no edges Add all the edges (E) to a set S and order them...
Let G -(V, E) be a graph. The complementary graph G of G has vertex set V. Two vertices are adjacent in G if and only if they are not adjacent in G. (a) For each of the following graphs, describe its complementary graph: (i) Km,.ni (i) W Are the resulting graphs connected? Justify your answers. (b) Describe the graph GUG. (c) If G is a simple graph with 15 edges and G has 13 edges, how many vertices does...