Graph Theory Definitions

Transcription

1 Grph Thory Dfinitions A grph is pir of sts (V, E) whr V is finit st ll th st of vrtis n E is st of 2-lmnt susts of V, ll th st of gs. W viw th gs s st of onntions twn th nos. Hr is n xmpl of grph G: G = (V, E) V = {,,,, } E = {{, }, {, }, {, }, {, }, {, }, {, }, {, }} Not tht this is th sm grph s th following (th st of vrtis is th sm, n th st of gs is th sm): In othr wors, whih wy you rw th grph is not importnt, it rmins th sm grph. W will oftn rw grphs without nming th vrtis, lik so, for xmpl: (Do you know wht this grph rprsnts?) Not: W sy on vrtx n mny vrtis (in othr wors, th plurl of vrtx is vrtis, n th singulr of vrtis is vrtx ). W somtims sy no inst of vrtx. Th plurl of no is simply nos. Not: By finition, mthmtil sts r unorr. This mns th st {, } is th sm s th st {, }, n so th gs of grph hv no irtion. If w wnt grph in whih th gs hv irtion w us irt grph, lso ll igrph. In igrph, th st of gs E is st of orr pirs of lmnts of V ; w writ n orr pir s (u, v) (this is iffrnt from th orr pir (v, u)). Hr is n xmpl of igrph: 1

2 G = (V, E) V = {,,,, } E = {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )} Not: Sts lso ignor rptition of lmnts (or on t llow rptitions ); tht is, th st {, } is th sm s th st {}. This hs two onsquns for th finition of grph: (1) sin gs r fin s 2-lmnt susts of V, w n hv no gs of th form {u, u} = {u} (us this is 1-lmnt sust of V ), n (2) w nnot rpt n g in th grph y listing it twi: y finition, th st of gs E = {{, }, {, }, {, }} is th sm s E = {{, }, {, }}. This mns tht our originl finition of grph os not llow ithr loops or multipl gs: loop multipl g Somtims, howvr, it is usful to llow loops n/or multipl gs, in whih s popl simply hng th finition of grph (us thy woul rthr hng th finition thn invnt nw wor). (Som popl us th trm simpl grph for grph without multipl gs, or for grph with nithr loops nor multipl gs, n you n sy looplss to mphsiz grph hs no loops; y fult, howvr, th lssil finition of grph os not llow loops or multipl gs.) On th othr hn, igrphs lwys llow loops, us (u, u) is vli orr pir of vrtis (th finition of orr pir os not rquir th two oorints of th orr pir to iffrnt). Not: In mny pls, you will s popl writ (u, v) for th g of grph vn in n unirt grph. Th nottion {u, v} is mor orrt, ut not s ommon, in ft. Dfinition 1 A wlk in grph G = (V, E) is squn of th form (v 1, {v 1, v 2 }, v 2, {v 2, v 3 }, v 3,...,v k, {v k, v k+1 }, v k+1 ) whr k 0, suh tht {v i, v i+1 } E for 1 i k. Th lngth of th wlk is k. For xmpl, hr is wlk in th first grph tht w show: wlk = {, {, },, {, },, {, },, {, }, } Not: Th finition llows wlks of lngth k = 0. Suh wlks onsist of singl vrtx: (v 1 ). Not: Th finition spifis whih gs r us y th wlk inst of simply spifying th squn of vrtis us y th wlk just in s th grph hs multipl gs (.g., if iffrnt finition of grph is ing us). Dfinition 2 A pth in grph G = (V, E) is wlk of lngth k 0 in whih th vrtis v 1,...,v k+1 r ll istint. 2

3 Th ov xmpl of wlk is not pth, sin it rpts th vrtx. Hr is pth: pth = {, {, },, {, },, {, },, {,}, } Not tht if wlk xists twn two vrtis of grph, thn pth lso xists. Dfinition 3 A yl in grph G = (V, E) is wlk of lngth k 1 in whih v 1,..., v k r istint n v 1 = v k+1. Hr is n xmpl: yl = {, {, },, {, },, {, },, {, },, {, }, } Not: If loops r llow, yl n onsist of singl loop. Dfinition 4 A grph G = (V, E) is onnt if thr is pth in G from u to v for vry u, v V. Dfinition 5 A onnt grph with no yls is tr. mphft: A tr on n vrtis hs n 1 gs, rgrlss of th shp th tr. Dfinition 6 A sugrph of grph G = (V, E) is grph G = (V, E ) suh tht V V, E E. Dfinition 7 An inu sugrph of grph G = (V, E) is sugrph G = (V, E ) suh tht ll gs of E with oth npoints in V r lso in E. G = G 1 = G 2 = non-inu surph inu sugrph Dfinition 8 A onnt omponnt (or simply omponnt) of grph G = (V, E) is mximl onnt inu sugrph of G. Not: Hr mximl mns tht th onnt sugrph is not ontin in ny lrgr onnt sugrph. Exmpl: Th following grph on 6 vrtis hs four onnt omponnts ( tringl n thr isolt vrtis): 3

4 G = Dfinition 9 A grph G is forst if h onnt omponnt of G is tr. Ft: A forst with n vrtis n m omponnts (nmly, m trs) hs n m gs. (Think of this: forst on 100 vrtis with two omponnts, on of 47 vrtis n th othr of 53 vrtis, hs = 98 vrtis, us tr on k vrtis hs k 1 gs.) Dfinition 10 Two vrtis of grph r jnt if n g joins thm. Dfinition 11 A liqu of grph G = (V, E) is st of vrtis V V suh tht ll pirs of vrtis in V r jnt. Dfinition 12 An inpnnt st of grph G = (V, E) is st of vrtis V V suh tht no vrtis in V r jnt. Dfinition 13 A vrtx n n g of grph r inint if th vrtx is n npoint of tht g. Also two gs of grph r inint if thy r inint to ommon vrtx (i.., shr n npoint). Two gs r isjoint if thy r not inint. Dfinition 14 A mthing of grph G = (V, E) is st of gs E E suh tht ll gs in E r isjoint. Exmpl (gs in th mthing r in ol): Dfinition 15 A grph G = (V, E) is iprtit if V is th union of two isjoint sts V 1 n V 2 suh tht vry g of G hs on npoint in V 1 n on npoint in V 2 (i.., suh tht V 1, V 2 r inpnnt sts). Dfinition 16 A ut of grph G = (V, E) is st of gs E E suh tht V is th isjoint union of two sts of vrtis V 1, V 2 suh tht vry g in E hs on npoint in V 1 n on npoint in V 2. Exmpl (nos in V 1 r lk, nos in V 2 r whit, ut gs r in ol): Dfinition 17 A hmiltonin pth of grph G is pth tht visits vry vrtx of G. Not: Hr visits on mns visits xtly on, y finition of pth. Hmiltonin pth xmpl: 4

5 Not: Not vry onnt grph hs hmiltonin pth. For xmpl, most trs on t hv hmiltonin pths. Dfinition 18 A hmiltonin yl of grph G is yl tht visits vry vrtx of G. Dfinition 19 A vrtx ovr of grph G = (V, E) is st of vrtis V V suh tht vry g in E is inint to vrtx in V. Exmpl (vrtx ovr nos r in lk): 5