12/9/14. CS151 Fall 20124Lecture (almost there) 12/6. Graphs. Seven Bridges of Königsberg. Leonard Euler

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1 CS5 Fll 04Leture (lmost there) /6 Seven Bridges of Königserg Grphs Prof. Tny Berger-Wolf Leonrd Euler Is it possile to wlk with route tht rosses eh ridge e Seven Bridges of Königserg Forget unimportnt detils. Forget even more. A vertex (or node, or point) A Grph e An edge (or line) So, wht is the Seven Bridges of Königserg prolem To find wlk tht visits eh edge extly one.

2 Question: Is it possile to find wlk tht visits eh edge e v e Suppose there is suh wlk, there is strting point nd n For every intermedite point v, there must e the sme num inoming nd outgoing edges, nd so v must hve n even num Question: Is it possile to find wlk tht visits eh edge e So, t most two verties n hve odd numer of edges. In this grph, every vertex hs only n odd e numer of edges, nd so there is no wlk Suppose there is suh wlk, there whih is visits strting eh point edge nd n extly one. For every intermedite point v, there must e the sme num inoming nd outgoing edges, nd so v must hve n even num So Euler showed tht the Seven Bridges of Königserg i When is it possile to hve wlk tht visits every edge ex So Euler showed tht the Seven Bridges of Königserg i When is it possile to hve wlk tht visits every edge ex Is it lwys possile to find suh wlk if there is t most two verties with odd numer of edges? Is it lwys possile to find suh wlk if there is NO! t most two verties with odd numer of edges?

3 So Euler showed tht the Seven Bridges of Königserg i When is it possile to hve wlk tht visits every edge ex So Euler showed tht the Seven Bridges of Königserg i When is it possile to hve wlk tht visits every edge ex Eulerin pth Euler s theorem: A grph hs n Eulerin pth if nd only if onneted nd hs t most two verties with n odd num Is it lwys possile to find suh wlk if the grph is onn YES! nd there re t most two verties with odd numer of edge This theorem ws proved in 736, nd ws regrded s the strting point of grph theor Types of Grphs A grph G=(V,E) onsists of: Simple Grphs Simple Grph Direted Grph A set of verties, V A set of undireted edges, E f V(G) = {,,,d,e,f} e E(G) = {d,f,d,e,d,e,df} G d Multi-Grph Eulerin pth prolem Two verties, re djent (neighours) if the edge 3

4 Vertex Degrees An edge uv is inident on the vertex u nd the vertex v. f The neighour set N(v) of vertex v is the set of verties djent to it. e.g. N() = {d,f}, N(d) = {,,,f}, N(e) = {,}. e degree of vertex = # of inident edges d e.g. deg(d) = 4, deg()=deg()=deg()=deg(e)=deg(f)=. Degree Sequene Is there grph with degree sequene (,,)? YES. Is there grph with degree sequene (3,3,3,3)? YES. Is there grph with degree sequene (,,)? NO. the degree of vertex v = the numer of neighours o For multigrphs, NO. For simple grphs, YES. Is there grph with degree sequene (,,,,)? NO. Wht s wrong with these sequenes? Where to go? Lemm. Hndshking Lemm For ny grph, sum of degrees = twie # edges E = deg( v) v V Lemm. Proof. Hndshking Lemm E = deg( v) v V Eh edge ontriutes to the sum on Q.E.D. the rig Corollry.. Sum of degree is n even numer.. Numer of odd degree verties is even. Question. Given degree sequene, if the sum of degree is it true tht there is grph with suh degree seq Exmples. ++ = odd, so impossile = odd, so impossile. For simple grphs, NO, onsider the degree sequene (3 For multigrphs (with self loops), YES! (esy y indut 4

5 Sme Grphs? Sme grph (different drwings) Sme grph (different lels) Alert Grnt Shrt Grph Isomorphism All tht mtters is the onnetions. Grphs with the sme onnetions re Informlly, isomorphi. two grphs re isomorphi if they re the sme G isomorphi to G mens there is n edge-preserving vertex mthing Sony Jessi Christos reltion preserving renming funtion Grph isomorphism hs pplitions like heking fingerprint, t 5