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QUESTION 13 Which of the following is a planar graph with 5 vertices and 4 faces? ) L j ) % ©) J\ (D) ‘§‘
QUESTION 16 (3 marks) Use the graph to identify whether each of the following is a cycle an open walk an open trail or a closed trail. D a) AFCFB [1 mark] b) AFCEFBA [1 mark] c) ABCDEFA [1 mark]
QUESTION 2 How many faces does this planar graph have? (A) 3 (B) 4 C) 5 (D) 7
QUESTION 6 This semi-Eulerian graph can be changed to an Eulerian graph by (A) adding a loop to vertex B. (B) removing the loop at vertex A. (C) adding an edge between vertices A and D. (D) removing the edge between vertices B and C.
QUESTION 21 (5 marks) The paths connecting various landmarks in a park are shown. D P Key B Bus stop C Coffee shop D Duck pond P Playground ¢ R Rose garden R W Water feature B a) Identify one cycle that passes through the rose garden and the playground. [1 mark] b) Identify whether the graph is Eulerian or semi-Eulerian. Justify your response. [2 marks] c) Construct an adjacency matrix from the graph using the vertex order listed in the key. [2 marks]
QUESTION 4 (5 marks) The table shows the current road length (in kilometres) between six towns. v [ vern [ o | e [ R | A T [ 6 s | [ % e | e | s e | e e | s w Cwew | T s e The government plans to build a direct road between Manon and Farra. Use a network diagram to determine the length of the direct road if it is to be 4 km shorter than the length of the current shortest road route between Manon and Farra.
QUESTION 20 (4 marks) Graphs 1| and 2 are shown. | 2 a) Show that Euler’s formula applies to graph 1. b) What feature in the drawing should be changed to represent graph 2 as planar? c) Redraw graph 2 to represent it as a planar graph. [2 marks] [1 mark] [1 mark]
QUESTION 5 (7 marks) At 9:00 am a security guard begins their patrol of the eight work sites represented in the network diagram starting and ending at site A. They drive at 40 km/h on the roads between sites and check every site once for 15 minutes. The length (km) of each road corresponds to the terms of the arithmetic sequence ¢ = ¢ + 2(n — 1) where ¢ = 1. Determine the earliest possible time the security guard can finish their patrol and identify the route they must follow. B ty C
QUESTION 9 The edges J R M P P Q on the network diagram above make (A) apath. (B) acycle. (C) aclosed trail. (D) an open walk.
QUESTION 14 Identify the planar graph that has four faces. (A) g (B) j (C) B (D) ;
QUESTION 25 (3 marks) The connected weighted graph below shows the time taken in minutes and seconds to walk between places in a park. Rose garden 1 min 55 sec 1 min 50 sec Butterfly house 3 min 10 sec 2 min 30 sec 2 min 20 sec 4 min 55 sec Tlayground Garden shed 2 min 20 sec Cafe a) Identify which two places are joined by a network bridge. [1 mark] b) Calculate the earliest time that a gardener could arrive at the butterfly house if they leave the garden shed at 8:55 am. [2 marks]
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