13. Let G = (V, E) be an undirected connected loop-free planar graph. Suppose G determines53 regions. If, for some planar embedding of G, each region has at least five edges in its boundary,prove that |V | > 81.
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13. Let G = (V, E) be an undirected connected loop-free planar graph. Suppose G determines
53 regions. If, for some planar embedding of G, each region has at least five edges in its boundary,
prove that |V | > 81.
Step by step
Solved in 2 steps
- Let u and v be distinct vertices in a connected graph G. There may be several connected subgraphs of G containing u and v. What is the minimum size of a connected subgraph of G containing u and v? Explain your answer.If G is a connected planar graph where e = 3v - 6, show that every region is triangular (has three boundary edges).Q₁. Construct a connected graph with at least 12 vertices such that the minimal dominating set has 6 vertices and the minimum dominating set has 4 vertices.
- Let G be a connected planar graph of order n > 3 and has no cycles of length 3. Prove that q≤2n4, where q is the number of edges in G.Let G = (V, E) be a connected graph with a bridge e = uv. Prove that there exist two disjoint sets of vertices U,W whose union is V where any path of G from vertices of U to vertices of W contains e.4. Prove: " Show that the number of edges in a simple planar graph of order n is at most 3n – 6". Give an example and explanation to validate the proof.
- Give an upper bound on the number e of edges of G in terms of n and g if G is a connected plane graph with n vertices and girth g.Let H be a connected planar graph with at least 3 vertices. Prove that f is greater than or equal to 2v-4, where v= number of vertices and f= faces of H. Show that for any integer v greater than or qual to 3, there exists a connected planar graph H that has v vertices and 2v-4 faces.In chess. A “knight’s move” consists of two squares either vertically or horizontally and then one square is a perpendicular direction. Depending on where the knight is situated, he has a minimum mobility of two moves—when in a corner—and a maximum mobility of eight moves. Let C be a graph with v=64, its vertices corresponding to the squares of a chessboard. Let two vertices of C be joined by an edge whenever a knight can go from one of the corresponding squares to the other in one move. Does C have an Euler Walk? Explain, but you do not have to draw C to answer.
- 3. An independent set of a graph G is a subset I of the vertex set V such that no two vertices in I are adjacent. Let i(G) be the size of a maximal independent set of G. (a) Show that I is an independent set of G if and only if V – I is a vertex cover of G. (b) Conclude from part (a) that i(G) + vc(G) = |V|.b,c,d