(7) Sketch any connected 3-regular Graph G with 6 vertices, determine how many edges must be...
Let G be a connected graph with n vertices and n edges. How many cycles does G have? Explain your answer.
how many edges does a 4-regular graph on n on vertices have?
Problem 1: In the graph below 6 5 4 1 3 (a) How many edges does the graph have? (b) Which vertices are odd, and which vertices are even? (c) is the graph connected? (d) Does the graph have any bridges? If so, list them all.
A forest contains 23 vertices and 20 edges. How many connected components does the graph have?
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...
er (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge ={a, b} is removed from then the subgraph S CG that remains is still connected. Directly' means using only the definitions of the concepts involved, in this case 'connected' and 'circuit'. Hint: If r and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y that avoids e? (b)...
Question 1# (a) Let G be a connected graph and C a non-trivial circuit in G. Prove directly that if an edge e fa, b is removed from C then the subgraph S C G that remains is still connected. "Directly' means using only the definitions of the concepts involved, in this case connected' and 'circuit'. Hint: If z and y are vertices of G connected by path that includes e, is there an alternative path connecting x to y...
Draw a simple undirected graph G that has 12 vertices, 18 edges, and 3 connected components. Why would it be impossible to draw G with 3 connected components if G has 66 edges?
2. Let G = (V, E) be an undirected connected graph with n vertices and with an edge-weight function w : E → Z. An edge (u, v) ∈ E is sparkling if it is contained in some minimum spanning tree (MST) of G. The computational problem is to return the set of all sparkling edges in E. Describe an efficient algorithm for this computational problem. You do not need to use pseudocode. What is the asymptotic time complexity of...
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...