By Fiorini, Stanley; Wilson, Robin J

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S F1 : .. .... .... ...... ... s s s s s G . G s ..................................................... ..... .. ...... ..... ... ... ... . ... . ... .. . . ... ... ..... . . . s F: s s ........ ............ ..... ...... .. ...... ...... ......... . .. . . ... ... . . ... .... ......... .. s............................................................................ .. ..............................................................

R8 r12 ........... 3. ..... sr7 ....... ... ... 4. s r8 ... ... 21. Modeling the art gallery by means of a graph. SOLUTION : (a) Let G be a graph of order 12 where V = {r1 , r2 , . . 21a). (b) If four guards are stationed in rooms r5 , r6 , r7 , r8 , then every room either contains a guard or is a neighboring room of some room containing a guard. 21b). This situation may suggest two other questions: (1) By placing the guards in rooms r5 , r6 , r7 , r8 , the eight rooms without guards are neighboring rooms of exactly one room with a guard.

19. Some graphs K r,r . 19 for r = 1, 2, 3. Notice that K 1,1 is K 2 , the graph K 2,2 is C 4 and K 3,3 is the graph we encountered when discussing the Three Houses and Three Utilities Problem in Chapter 1. SUBGRAPHS While many problems in graph theory deal with degrees of vertices, there are many others that concern the structure of graphs that lie within a given graph. A graph H is called a subgraph of a graph G if every vertex and edge of H is a vertex and edge, respectively, of G.

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