a g f 8 e 11 Also: Minimum degree, maximum degree, vertex of degree d 1 adjacent to vertex of degree d 2,...

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1 Warmup: Lt b 2 c 3 d a 8 11 (i) Vriy tat G is connctd by ivin an xampl o a walk rom vrtx a to ac o t vrtics b. (ii) Wat is t sortst pat rom a to c? to? (iii) Wat is t lonst pat rom a to c? to? (iv) Wat is a lonst pat in G? (v) Dos G av any maximal pats tat ar sortr tan t pat in part (iv)? (Rcall a pat is a walk wit no rpatd vrtics or ds, and a maximal pat is on tat can t b xtndd in itr dirction.) Grap invariants Rcall, a rap invariant is a statistic about a rap tat is prsrvd undr isomorpisms (rlablin o t vrtics). Namly, i you don t nd t labls to calculat t statistic, tn it s probably a rap invariant. 1. V, E 2. Dr squnc Also: Minimum dr, maximum dr, vrtx o dr d 1 adjacnt to vrtx o dr d 2, Bipartit or not I any subrap is not bipartit, tn G is not bipartit. A rap is bipartit i and only i it as no odd cycls as subraps. 4. Pats or cycls o particular lnts Also: lonst pat or cycl lnt, maximal pats o crtain lnts,... You try: ICE 40(a)

2 How connctd? Suppos G is connctd ow can w valuat ow wll connctd it is? For xampl, i you r buildin a ntwork o computrs, can your ntwork b disconnctd i on computr or on lin ails? I t subrap G v is not connctd, w call v a cut vrtx. Similarly, i G is not connctd, w call a cut d. For xampl, in b c d a is a cut vrtx. G dosn t av any cut ds. How connctd? I t subrap G v is not connctd, w call v a cut vrtx. Similarly, i G is not connctd, w call a cut d. For xampl, in b c d a is a cut vrtx. G dosn t av any cut ds. In b c d H a t cut vrtics ar b and, and d a b is t only cut d.

3 You try: Idntiy t cut ds and vrtics (i any) o t ollowin raps: b c d H a K I W Ä V as t proprty tat GrV W s is not connctd, w call W a vrtx cut. IF Ä E as t proprty tat G E is not connctd, w say F is an d cut. For xampl, in K on xampl o a vrtx cut is tc, d, u. Anotr vrtx cut is tb,, u. On xampl o an d cut is tb c, b d, c,, u. Anotr d cut is tc, d, u. CAREFUL! In cut vrtx, vrtx is t noun and cut is t adjctiv; in vrtx cut, cut is t noun and vrtx is t adjctiv. Sam tin in cut d vrsus d cut.

4 Vrtx connctivity Lt applpgq b t wst numbr o vrtics ndd to disconnct a rap (or to wittl it down to a sinl vrtx, wicvr is wr). W call applpgq t (vrtx) connctivity o G. How to comput: To sow applpgq k, you av to iv a vrtx cut o siz k and sow tat rmovin all possibl substs o siz k lavs a connctd raps. applpgq 3 Not tat t complt rap as no vrtx cut, so w din applpk n q n 1. Tus 0 applpgq V 1. T larr t appl, t mor connctd t rap. W say G is k-connctd i applpgq k. Ed connctivity Lt pgq b t wst numbr o ds ndd to disconnct a rap. W call pgq t d connctivity o G. How to comput: To sow pgq `, you av to iv an d cut o siz ` and sow tat rmovin all possibl substs o siz ` lavs a connctd raps. pgq 3 Not tat w can disconnct a rap by rmovin all t ds around a sinl vrtx. So pgq min dpvq. vpv Morovr, i w rmov all t vrtics adjacnt to v, tnv is isolatd. So applpgq pgq min pdpvqq. vpv

5 applpgq : pgq : min pdpvqq : vpv applpgq : H pgq : min pdpvqq : vpv Grap invariants Rcall, a rap invariant is a statistic about a rap tat is prsrvd undr isomorpisms (rlablin o t vrtics). Namly, i you don t nd t labls to calculat t statistic, tn it s probably a rap invariant. 1. V, E 2. Dr squnc Also: Minimum dr, maximum dr, vrtx o dr d 1 adjacnt to vrtx o dr d 2, Bipartit or not I any subrap is not bipartit, tn G is not bipartit. A rap is bipartit i and only i it as no odd cycls as subraps. 4. Pats or cycls o particular lnts Also: lonst pat or cycl lnt, maximal pats o crtain lnts, Ed and vrtx connctivity You try: 40(b) ().

6 Eulrian trails and circuits Suppos you r tryin to dsin a maximally cint rout or postal dlivry, or strt clanin. You want walk on t city strts tat visits vry strt xactly onc. T Svn Brids o Könisbr, Lonard Eulr (1736) C A D B Qustion: is it possibl to start at som location in t town, travl across all t brids onc witout crossin any brid twic? Eulrian trails and circuits Wat Eulr did was modl t problm as t multirap C C A D A D B In onor o is contribution, w say tat an Eulrian trail in a rap G is a trail (no rpatd ds) tat passs trou vry d o G (xactly onc). An Eulrian circuit is an Eulrian trail tat nds wr it startd. Ncssary: Connctd; at most two vrtics o odd dr. Tis is also a su cint condition. Wy?... B

7 Eulrian trails and circuits An Eulrian trail in a rap G is a trail (no rpatd ds) tat passs trou vry d o G (xactly onc). An Eulrian circuit is an Eulrian trail tat nds wr it startd. Ncssary: Connctd; at most two vrtics o odd dr. Tis is also a su cint condition. Wy?... Aloritm or indin an Eulrian circuit in any rap wit all vn dr vrtics: Start anywr and o until you t stuck you ll b back wr you startd. Somwr in t middl, you av a vrtx wr you didn t xaust t ds incidnt. Go back and start rom tr and o until you t stuck. Rpat. b d a c Eulrian trails and circuits An Eulrian trail in a rap G is a trail (no rpatd ds) tat passs trou vry d o G (xactly onc). An Eulrian circuit is an Eulrian trail tat nds wr it startd. Ncssary: Connctd; at most two vrtics o odd dr. Tis is also a su cint condition. Wy?... Aloritm or indin an Eulrian trail in any rap wit all but two vn dr vrtics: Start at an odd-dr vrtx and o until you t stuck. Somwr in t middl, you av a vrtx wr you didn t xaust t ds incidnt. Go back and start rom tr and o until you t stuck. Rpat.

8 Eulrian trails and circuits Torm A rap as an Eulrian trail i and only i it is connctd and as at most two vrtics o odd dr. Furtr, a connctd rap as an Eulrian circuit i and only i vry vrtx is o vn dr.