CIT 596 Theory of Computation 1. Graphs and Digraphs

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1 CIT 596 Theory of Computtion 1 A grph G = (V (G), E(G)) onsists of two finite sets: V (G), the vertex set of the grph, often enote y just V, whih is nonempty set of elements lle verties, n E(G), the ege set of the grph, often enote y just E, whih is possily empty set of elements lle eges, suh tht eh ege e in E is ssigne n unorere pir (u, v) of verties, lle the en verties of e. Sometimes, it is onvenient to enote (u, v) y simply uv, or equivlently, vu.

2 CIT 596 Theory of Computtion 2 Consier the grph G = (V, E) suh tht V = {,,,, e} n E = {e 1, e 2, e 3, e 4, e 5, e 6, e 7, e 8 }, where e 1 (, ) e 2 (, ) e 3 (, ) e 4 (, ) e 5 (, ) e 6 (, e) e 7 (, e) e 8 (, e)

3 CIT 596 Theory of Computtion 3 A grph is often represente y igrm in whih verties re rwn s irles n eges s line or urve segments joining the irles representing the en verties of the ege. e 8 e 1 e e 7 e 2 e 5 e 6 e 3 e 4

4 CIT 596 Theory of Computtion 4 Verties re lso lle points, noes, or just ots. If e is n ege with en verties u n v then e is si to join u n v. Note tht the efinition of grph llows the possiility of the ege e hving ietil en verties, i.e., it is possile to hve vertex u joine to itself y n ege suh n ege is lle loop. If two (or more) eges hve the sme en verties then eges re lle prllel. A grph is lle simple is it hs no loops n no prllel eges.

5 CIT 596 Theory of Computtion 5 For n exmple of simple grph, onsier the grph G = (V, E) suh tht V = {,,, } n E = {e 1, e 2, e 3, e 4 }, where e 1 (, ) e 2 (, ) e 3 (, ) e 4 (, ) Some uthors use the term multigrph for grphs with loops n prllel eges, n reserve the term grph for simple grphs only. Sine we will el with grphs with loops very often, it is more onvenient not to mke this istintion.

6 CIT 596 Theory of Computtion 6 A pitoril representtion of the simple grph G in the previous slie: e 1 e 2 e 3 e 4

7 CIT 596 Theory of Computtion 7 The numer of verties in G is lle the orer of G. The numer of eges in G is lle the size of G. Two verties u n v of grph G re si to e jent if uv E(G). If uv E(G) then we sy tht u n v re non-jent verties. An ege e of grph G is si to e inient with or inient to the vertex v if v is n en vertex of e. In this se, we lso sy tht v is inient with or inient to e. Two eges e n f whih re inient with ommon vertex v re si to e jent.

8 CIT 596 Theory of Computtion 8 Let v e vertex of the grph G. The egree (v) of v is the numer of eges of G inient to v, ounting eh loop twie, i.e., it is the numer of times v is n en vertex of n ege. e For exmple, () = 1, () = 3, () = 3, () = 3, n (e) = 4 in the grph ove.

9 CIT 596 Theory of Computtion 9 The First Theorem of Grph Theory. For ny grph G with n e eges n n v verties v 1,..., v nv, we hve tht Chek y yourself: n v i=1 (v i ) = 2 n e. e

10 CIT 596 Theory of Computtion 10 A wlk in grph G is finite sequene W = v 0 e 1 v 1 e 2 v 2... v k 1 e k v k whose terms re lterntely verties n eges suh tht, for 1 i k, the ege e i hs en verties v i 1 n v i. We sy tht the ove wlk is v 0 v k wlk or wlk from v 0 to v k. The integer k, the numer of eges of the wlk, is lle the length of W. A trivil wlk is one ontining no eges.

11 CIT 596 Theory of Computtion 11 The sequene e 2 e 8 f e 6 e 3 e 4 e e 4 is wlk of length 6 in the grph elow. e 1 e 2 e 3 e 8 e 4 e 7 e 6 e 5 f e In simple grph, wlk is etermine y the sequene of verties only: f e.

12 CIT 596 Theory of Computtion 12 Given two verties u n v of grph G, u v wlk is lle lose or open epening on whether u = v or u v. If the eges e 1, e 2,..., e k of the wlk v 0 e 1 v 1 e 2 v 2... v k 1 e k v k re istint then W is lle tril. If the verties v 0, v 1,..., v k of the wlk v 0 e 1 v 1 e 2 v 2... v k 1 e k v k re istint then W is lle pth. For the grph in the previous slie, f e is f tril, ut not pth! The sequene f e is f e pth.

13 CIT 596 Theory of Computtion 13 A vertex u is si to e onnete to vertex v in grph G if there is pth in G from u to v. A grph G is onnete if every two verties of G re onnete; otherwise, G is isonnete. M N Connete Disonnete

14 CIT 596 Theory of Computtion 14 A nontrivil lose tril C = v 1 v 2... v n v 1 in grph G is lle yle if the verties v 2... v n re ll istint. A yle of length k, i.e., yle with k eges, is lle k-yle. For exmple, f is 3-yle in the grph elow: e 1 e 2 e 3 e 8 e 4 e 7 e 6 e 5 f e

15 CIT 596 Theory of Computtion 15 A grph G is si to e yli if it ontins no yles. A grph G is lle tree if it is onnete n yli. e The verties of egree (t most) 1 in tree re lle the leves of the tree.

16 CIT 596 Theory of Computtion 16 A irete grph (or simply igrph) D = (V (D), A(D)) onsists of two finite sets: V (D), the vertex set of the igrph, often enote y just V, whih is nonempty set of elements lle verties, n A(D), the r set of the igrph, often enote y just A, whih is possily empty set of elements lle rs, suh tht eh r in A is ssigne (orere) pir (u, v) of verties. If is n r in D with ssoite orere pir of verties (u, v), then is si to join u to v, u is lle the initil vertex of, n v is lle the terminl vertex of.

17 CIT 596 Theory of Computtion 17 For exmple, onsier the igrph D = (V (D), A(D)) suh tht V (D) = {,,,, e} n A(D) = {e 1, e 2, e 3, e 4, e 5, e 6 }, suh tht e 1 (, ), e 2 (, ), e 3 (, ), e 3 (, e), e 4 (e, ), n e 6 (e, e). e 6 e e 1 e 5 e 4 e 2 e 3

18 CIT 596 Theory of Computtion 18 Let D e igrph. Then irete wlk in D is finite sequene W = v 0 1 v 1... k v k, whose terms re lterntely verties n rs suh tht for 1 i k, the initil vertex of the r i is v i 1 n its terminl vertex is v i. The numer k of rs is the length of W. The wlk W given in the efinition ove is si to e v 0 v k irete wlk or irete wlk from v 0 to v k. There re similr efinitions for irete trils, irete pths, n irete yles.

19 CIT 596 Theory of Computtion 19 For exmple, onsier the igrph D elow: e 6 e e 1 e 5 e 4 e 2 e 3 The sequene e 1 e 3 e 4 e is e irete wlk, e irete tril, n e irete pth in D.

20 CIT 596 Theory of Computtion 20 A vertex v of the igrph D is si to e rehle from vertex u if there is irete pth from u to v. Given ny igrph D = (V (D), A(D)), we n otin grph G = (V (G), E(G)) from D s follows: Let V (G) = V (D) n E(G) = {e (, ) (, ) A(D)}. The grph G is the unerlying grph of D. A igrph D is si to e wekly onnete (or simply onnete) if its unerlying grph is onnete. A igrph D is si to e strongly onnete (or ionnete) if for ny pir of verties u n v in D there is irete pth from u to v.

21 CIT 596 Theory of Computtion 21 e 6 e 6 e e e 1 e 5 e 4 e 5 e 1 e 4 e 2 e 2 e 3 e 3 A igrph Its unerlying grph Is the ove irete grph onnete? If so, is it strongly onnete?

22 CIT 596 Theory of Computtion 22 Let v e vertex in igrph D. The inegree i(v) of v is the numer of rs of D tht hve v s the terminl vertex, i.e., the numer of rs tht go to to v. Similrly, the outegree o(v) of v is the numer of rs of D tht hve v s the initil vertex, i.e., the numer of rs tht go out of v. The First Theorem of Digrph Theory. Let D e igrph with n verties n q rs. If v 1,..., v n is the set of verties of D, then we hve tht n i=1 i(v i ) = n i=1 Cn you figure out the proof for this one? o(v i ) = q.