D math graph

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  • 1. 7.1. Preliminaries &Graph Terminology Reference: A Course in Discrete Mathematical Structures by LR Vermani Budi Irmawati-NAISTs student

2. 2IsomorphicG (V,E) G(V,E)x (x,y) =VV x =(x)y =EE y =(y)(x,y) = (x,y)zz =(z)4 3 d cc d1 2 ab ab 3. 3MultigraphGraph that has multipleedges connecting the sameverticesDirected MultigraphDirected graph that ismultigraphSimple/Linear Graphgraph which no loop and onlyhave one edge between twovertices 4. 4Weighted graphGraph which each of itsedges are weighted.Ordered quadruple (V,E,f,g)or triple (V,E,f) or triple (V,E,f)V : non-empty set of verticesE : set/multiset of edgesf : function with domain Vg : map with domain E 5. 5Bipartite VGraphV1 V2 sub graph 1 (male) (female) Each vertex in V1 incident with an vertex in V2 a1 b1 a2 b2 an bn V sub graph 2 Every vertex in V1 may not V1 V2 incident with a vertex in V2 (male) (female) Some vertex in V1 or V2 are a1 b1 isolateddisjoint a2 b2disjoint an bn 6. 6BipartiteGraph Fig. 7.33V1 = {a,b,g}V2 = {c,d,e,f}Fig. 7.34V = V1V2V1V2 =V1V2not bipartite 7. 7BipartiteComplete BipartiteGraph GraphV = V1 V2V1 V2 =V1V2Every vertex in V1 adjacentevery vertex in V2 and viceversaKm,nComplete bipartite graphwhich V1 order m and V2order nThe number of edges is mninstead of n(n-1)/2Every vertex adjacent toevery vertex in othersubgraph 8. 8Bipartite Graph simple graph: two vertices one edgesimple graph: at least three vertices (possibly) always bipartite trivial graph; three isolated vertices only have one edge always have two edges, orbipartite have three edges 9. 9Example 7.2Cube with 6 faces. Every cube are colored using 4 colors.Is it possible to stack the cubes to form column so that no colorapears twice 10. 10 1 2 3 4a R G W Bb G W B Rc W R B G Example 7.2d G B W RCube with 6 faces. Every cube are colored using 4 colors. Is it possible to stack the cubes to form column so that no color apears twice 11. 11 Possible crossingone persona couplethree ladiesthree gentscouple & gentExample 7.3How to cross a river for:5 couples1 boat for 3 personswife + husband if there a man 12. 12Example 7.4How to cross a river for:5 couples1 boat for at most 4 personswife + husband if there a man 13. 13Example 7.5A man, a dog, a sheep, a basket of cabbageOnly can carry one item in crossing a riverCabbage cannot stay with a sheep and a dog cannot stay with a sheepHow to cross it ? 14. 14Example 7.6G = (V,E) is undirected graph with k componentso(V) = n, o(E) = mProve that m n - k G = (V,E) where o(V) = s 2.Vertex v1. G is connected graph. v1 adjacent to v2, v2 to v3, and so on.Chain v1v2, v2v3, , vs-1vs. at least s-1 edges in GFor o(V) = 1, o(E) o(V) 1 15. 15 Example 7.6 For k components, (E1,V1), (E2,V2), , (Ek,Vk) EiEj =for i j E = E1 E2 Ek and V = V1 V2 Vk o(E) = o(E1)+o(E2)++o(Ek) (o(V1)-1)+(o(V2)-1)++(o(Vk)-1))= o(v1)+o(V2)++o(Vk) k= o(V) - k