Matchings (Bipartite Graphs) Decision Maths Unit 6 Matchings (Bipartite Graphs)

As none of the girls want to be attached to Dave, a complete match is not possible

Group X Group Y A R B W C G D P

and if possible a complete match.

5

YOU MUST DISPLAY THIS FINAL MATCH

Initial Match E to 3 Initial Match E to 5 A 1 A 1 B 2 B 2 C 3 C 3 D 4 Complete Match (A – 1) (B – 4) (C – 2) (D – 5) (E – 3) Complete Match (A – 1) (B – 4) (C – 3) (D – 2) (E – 5)

Group X Group Y Group X Group Y A M A M D P D P E N E N J S S J Original Bipartite Graph Initial Match

Group X Group Y Group X Group Y A M A M D P D P E N E N S J J S Original Bipartite Graph Complete match starting from E to P E – P – D – N (A,M) (D,N) (E,P) (J,S)

Group X Group Y Group X Group Y A M A M D P D P E N E N J S S J Original Bipartite Graph Complete match starting from E to M E – M – A – P – D – N (A,P) (D,N) (E,M) (J,S) Both of these solutions are equally valid

Adjacency matrix A – 5 ┼ B - 3 ┼ F – 1 - 2 ┼ D – 6 C – 4 ┼ E 1 1 1 1 1 1 1 1 1 1 1 1 1 A – 5 ┼ B - 3 ┼ F – 1 X Y C – 4 ┼ E - 2 ┼ D – 6 A 1 Final Match (A ,5) (B,3) (C,4) (D,6) (E,2) (F,1) B 2 C 3 D 4 E 5 F 6

Matchings By the end of the topic you should know how to Draw a Bipartite graph from information given as MATCHINGS or in an ADJACENCY TABLE Draw an ADJACENCY TABLE from information given in a bipartite graph. From an INITIAL PAIRINGS draw a bipartite graph that shows a complete match Show the correct notation for each match, e.g. A – 3 ┼ E etc. State the completed matched pairings as a FINAL MATCH For example (A,3) (B,4) (C,1) (D,5) (E,2)

BIPARTITE GRAPHS If you have to draw a bipartite graph you must show, if asked for The graph with all the matchings The graph with the INITIAL matchings Another graph to show the COMPLETE matchings, with deletions (although this is not marked) Show the notation for each matching (This is marked) A – 3 ┼ C – 1 ┼ B etc. State the completed matched pairings. (This is marked) For example (A – 3) (B – 4) (C – 1) (D – 5) (E – 2)