OLD AND NEW GENERALIZATIONS OF LINE GRAPHS

Transcription

1 IJMMS 2004:29, PII. S Hindawi Publishing Cop. OLD AND NEW GENERALIZATIONS OF LINE GRAPHS JAY BAGGA Received 8 Octobe 2003 Line gaphs have been studied fo ove seventy yeas. In 1932, H. Whitney showed that fo connected gaphs, edge-isomophism implies isomophism except fo K 3 and K 1,3.The line gaph tansfomation is one of the most widely studied of all gaph tansfomations. In its long histoy, the concept has been ediscoveed seveal times, with diffeent names such as deived gaph, intechange gaph, and edge-to-vetex dual. Line gaphs can also be consideed as intesection gaphs. Seveal vaiations and genealizations of line gaphs have been poposed and studied. These include the concepts of total gaphs, path gaphs, and othes. In this bief suvey we descibe these and some moe ecent genealizations and extensions including supe line gaphs and tiangle gaphs Mathematics Subject Classification: 05C75, 05C45, 05C62, 05C Intoduction. The line gaph L(G) of a gaph G is defined to have as its vetices the edges of G, with two being adjacent if the coesponding edges shae a vetex in G. Line gaphs have a ich histoy. The name line gaph was fist used by Haay and Noman [17] in But line gaphs wee the subject of investigation as fa back as 1932 in Whitney s pape [24], whee he studied edge isomophism and showed that fo connected gaphs, edge-isomophism implies isomophism except fo K 3 and K 1,3.The fist chaacteization (patition into complete subgaphs) was given by Kausz [19]. Since this is a suvey on genealizations of line gaphs, we will not descibe line gaphs and thei popeties in any detail hee. Instead, we efe the inteested eade to a somewhat olde but still an excellent suvey on line gaphs and line digaphs by Hemminge and Beineke [18]. A moe ecent book by Pisne [22] descibes many inteesting genealizations of line gaphs. Fo geneal gaph theoetic concepts and teminology not defined hee, please see [9, 16]. In the est of this section, we will make some geneal emaks on the natue of eseach in line gaphs and the genealizations that have been studied. In the following sections, we descibe some genealizations and vaiations of the line gaph concept. We descibe two, supe line gaphs and tiangle gaphs, in somewhat geate detail. We also mention some open poblems in this aea. Accoding to the above-mentioned aticle by Hemminge and Beineke, much of the effot in the ealy eseach concentated on the detemination poblem (detemine which gaphs have a given gaph as thei line gaph) and the chaacteization poblem (chaacteize those gaphs that ae line gaphs of some gaph). Kausz [19] gave a chaacteization of line gaphs in tems of complete subgaphs. Thee ae seveal othe chaacteizations mentioned in [18], including the fobidden subgaph chaacteization

2 1510 JAY BAGGA Figue 1.1 by Beineke [11], which states that a gaph is a line gaph if and only if it has no induced subgaph isomophic to any of the gaphs shown in Figue 1.1. One can view the line gaph as a tansfomation G L(G). Repeated applications of this tansfomation yield iteated line gaphs, which have been studied by seveal authos. Thus iteated line gaphs ae defined by L 1 (G) = L(G) and L n+1 (G) = L(L n (G)) (povided that L n (G) is not null). Questions about convegence have been consideed. If G is the path P n on n vetices, then L(G) = P n 1.Hence{L n } teminates fo paths. If G is a cycle, then L(G) G. AlsoL(K 1,3 ) = K 3. Hence fo these gaphs {L n } becomes a constant. Fo all othe connected gaphs, {L n } contains abitaily lage gaphs [23]. Othe popeties of iteated line gaphs such as Hamiltonicity have also been studied. Theoem 1.1 [13]. Let G be a connected gaph of ode p that is not a path. Then L n (G) is Hamiltonian fo all n p 3. See Pisne [22] fo details on the line gaph tansfomation and seveal othe simila concepts and esults on iteations. Line gaphs have edges as thei vetices. An edge can be viewed as a subset of two adjacent vetices, a clique of ode two, a path of length one, o a subgaph of size one, among othes. Depending on which view is consideed, genealizations and extensions of line gaphs esult by genealizing the coesponding view. Fo example, in Section 4, we will descibe a ecent genealization called the tiangle gaph fo which the vetex set consists of the tiangles of G, with two being adjacent if they shae an edge. In the next section, we descibe some ealie genealizations of line gaphs. 2. Old genealizations 2.1. Total gaphs. The total gaph T(G) of a gaph G, defined by Behzad and Chatand [10], takes both the vetices and edges of G as elements fo its set of vetices.

3 OLD AND NEW GENERALIZATIONS OF LINE GRAPHS 1511 u w uvw v x uvx vxy xyz wvx vxz xzy y z yxz G P 2 (G) Figue 2.1 Two vetices ae adjacent if the coesponding elements of G ae eithe adjacent o incident. Recall Vizing s theoem, which states that if G has maximum degee, then χ(l(g))= o + 1, Behzad conjectued that χ(t(g)) = + 1o + 2. Whethe any gaph has total chomatic numbe geate than +2 is still an open question [15] Middle gaphs. The middle gaph mid(g) of G is obtained fom G by inseting a new vetex into evey edge of G and by joining those pais of new vetices which lie on adjacent edges of G. Thus mid(g) can also be consideed as the intesection gaph of all K 1 s and K 2 s in G [22]. We obseve that (i) L(G) is an induced subgaph of mid(g), (ii) the subdivision gaph S(G) is a subgaph of mid(g), (iii) mid(g) is a subgaph of T(G) Entie gaphs. Theentiegaphe(G) of a plane gaph G is the gaph whose vetices coespond to vetices, edges, and egions of G. Two vetices of e(g) ae adjacent if the coesponding elements of G ae adjacent o incident. See [20] fo some esults on entie gaphs Path gaphs. The thpath gaph (G) [12] has the paths of length in G as its vetices and two such paths ae adjacent if thei union is a path o cycle of length +1. Fo = 1, we obtain the line gaph. An example fo = 2 is shown in Figue 2.1. We obseve that a path of length in G coesponds to an isolated vetex in (G) if and only if its end vetices have degee 1 in G. Thus, connectedness of G is not inheited by the th path gaph fo >1. Boesma and Hoede [12] also show that the numbes of vetices and edges in 2 (G) ae ( ) deg(v) v and (1/2) 2 v[(deg(v) 1) u v(deg(u) 1)], espectively (whee denotes adjacency) subgaph distance gaphs. Let (G) be the gaph whose vetices ae the subgaphs of G having edges, with two such being adjacent if the symmetic diffeence of thei edge sets consists pecisely of two adjacent edges of G. S (G) is called

4 1512 JAY BAGGA a b ab c d ac ad bc bd cd G S 2 (G) Figue 2.2 the -subgaph distance gaph (Chatand et al. [14]). An example fo = 2 is shown in Figue 2.2. Note that two subgaphs (with edges) in G ae adjacent if one can be obtained fom the othe by pivoting one edge. Because (G) is connected wheneve G is connected (and has at least edges), it povides a measue of the distance between two subgaphs of size in G. 3. Supe line gaphs. In this section, we descibe in somewhat geate detail a moe ecent genealization, the supe line gaph. Supe line gaphs wee intoduced by Bagga et al. in [6]. Since then, the study of supe line gaphs has pogessed much futhe [1, 2, 3, 4, 5, 6] Definition and basic popeties. Fo a fixed intege (with 1 q = E(G) ), the supe line gaph (G) of index has the sets of edges in G as its vetices, and two vetices ae adjacent if an edge in one set is adjacent (in G) to an edge in the othe. It follows that 1 (G) is the usual line gaph. Figue 3.1 shows an example of a gaph G and the gaph 2 (G). Fo simplicity, we denote a set {x,y} of edges by xy. Seveal vaiations of the definition of a supe line gaph can be consideed. Fo example, one could fom a multigaph by joining two vetices with as many edges as thee ae adjacencies between the two sets of edges. We call this the supe line multigaph. See [3] fo some esults. O, we can fom the intesection gaph of the sets of vetices on the two sets of edges. The supe line gaph opeato has nice heeditay popeties, as the next two esults show [1]. Theoem 3.1. (H). If G is a subgaph of gaph H, then (G) is an induced subgaph of Theoem 3.2. Let G be a gaph with q edges. Fo <q/2, (G) is isomophic to a subgaph of +1 (G). The definition of adjacency in the supe line gaph (G) implies that if two vetices (say) S and T ae nonadjacent, then (with S and T consideed as -sets of edges) thei

5 OLD AND NEW GENERALIZATIONS OF LINE GRAPHS 1513 b a c ab bc G : L 2 (G) : ac bd d ad cd Figue 3.1 intesection S T is a set of isolated edges in the subgaph (of G) of thei union S T. This leads to ou next esult [6]. Theoem 3.3. If S and T ae -sets of edges in G such that neithe consists entiely of isolated components of G, then the distance between S and T in (G) is 1 o 2. Poof. If such S and T ae not adjacent in (G),thenGcontains an edge e adjacent to some edge in S and an edge f adjacent to some edge in T.NowletR be any set of edges of G containing e and f.thenboths and T ae adjacent to R in (G), andso the distance between them is 2. Coollay 3.4. If G is a gaph in which fewe than components ae isolated edges, then (G) has diamete 1 o 2. Coollay 3.5. Fo a gaph G, at most one component of (G) is nontivial Line completion numbe. We obseve that if (G) is complete, so is s (G) fo s q. This leads us to define the line completion numbe lc(g) of gaph G to be the minimum index fo which (G) is complete. Clealy, lc(g) q. We obseve that the only gaphs with the line completion numbe 1 ae the stas and K 3. A geneal bound on line completion numbe is given by the following esult [6]. Theoem 3.6. If G is a gaph with q edges and c components, then lc(g) (q + c)/2, and this bound is shap. The next theoem [5] descibes which gaphs have small o lage line completion numbes. Theoem 3.7. (i) lc(g) = 1 if and only if G is K 1,n o K 3. (ii) lc(g) 2 if and only if G does not have 3K 2 o 2P 3 as a subgaph. (iii) lc(g) 3 if and only if G does not have any of the following as a subgaph: 4K 2, K 2 2K 1,2, 2K 3, K 3 P 4, K 3 K 1,3, 2K 1,3, K 1,3 P 4. (iv) lc(g) = q if and only if G qk 2. (v) lc(g) = q 1 if and only if G P 3 (q 2)K 2 o G 2P 3 (q 4)K 2. Line completion numbes of seveal classes of gaphs have been found. We list some of these in the next theoem. See [5] fo moe details.

6 1514 JAY BAGGA Theoem 3.8. (i) If T is a tee of ode n 2, then lc(t ) n/2. Futhemoe, fo any intege k satisfying 1 k n/2, thee is a tee T of ode n with lc(t ) = k. (ii) lc(k n ) = (1/2) n/2 ( n/2 1)+1. (iii) lc(p n ) = lc(c n ) = n/2. (iv) lc(k 1 +P n ) = lc(k 1 +C n ) = 2n/3. (v) lc(k 1 +nk 2 ) = 3n/4 +1. Line completion numbes of seveal othe classes of gaphs have also been obtained. These classes include hypecubes, laddes, and gids. One class of gaphs fo which only patial esults ae known is the class of complete bipatite gaphs. We list the known cases in the next theoem. Theoem 3.9. (i) Fo m = 2, n = 2s, lc(k m,n ) = s+1 = mn/4+1. (ii) If m n, and m is odd, then lc(k m,n ) = n( 2 1)/4m. (iii) lc(k 2,2q+1 ) = 2 q. (iv) lc(k 2,2q+3 ) = (q+1) fo q 1. (v) lc(k 2 +1,2 +4k ) = ( +k) 2 fo 4k 2 3k. A complete detemination of lc(k m,n ) is still open Cycles in supe line gaphs. We now descibe seveal inteesting esults on cycles in supe line gaphs. Even though supe line gaphs can be dense, they, in geneal, do not satisfy well-known sufficient conditions fo Hamiltonicity. Fo any connected gaph G, howeve, 2 (G) tuns out to be vetex pancyclic, that is, evey vetex lies on cycles of length thee though the ode of the gaph. In fact, as we see below, this esult is tue fo many disconnected gaphs also. We will state most esults without poof. Howeve, to give a flavo of the techniques used, we include a special esult with poof. Theoem If T is a tee with q 2 edges, then 2 (T ) is pancyclic. Poof. The poof is by induction on q.foq 4, it can be easily checked that 2 (T ) is complete, which is also tue if T is a sta. Assume that T is not a sta and that q>4. Choose a vetex u of T such that deg(u) > 1, and evey neighbo of u, exceptone, say v, is an end vetex. Since T is not a sta, such a vetex always exists. Let S be the sta with cente u, and let T v be the subtee of T obtained by emoving u and all its neighbos except v. Lets = deg(u) so that T v has = q s edges. By ou choice of u and v, it follows that >0 and s>1. If = 1, then 2 (T ) is complete. If = 2 with (say) E(T v ) ={f 1,f 2 },then 2 (T ) f 1 f 2 is complete, and deg 2 (T ) f 1 f 2 > 1. Hence 2 (T ) is pancyclic. Thus we may assume that 3. Now the vetex set of 2 (T ) is the (disjoint) union of V( 2 (S)), V( 2 (T v )), andthe set W ={ab a E(T v ), b E(S)}. Also, 2 (S) is complete and, by the induction hypothesis, 2 (T v ) is pancyclic. Let C v be a spanning cycle of 2 (T v ).AlsoletX 1 X 2 be an edge of C v, whee X 1 = f 1 e and X 2 = f 2 f, whee f 1,e,f 2,f E(T v ) and f 1 is adjacent to f 2. We obseve that W =s.sinces is a sta with s edges, the subgaph of 2 (T ) induced by W contains the complete multipatite gaph K,,..., as a spanning subgaph.

7 OLD AND NEW GENERALIZATIONS OF LINE GRAPHS 1515 Hence, it contains paths of lengths 1 though s 1. We can also assume that the end vetices of each of these paths ae Y 1 = f 1 g and Y 2 = f 2 h, whee f 1 and f 2 ae as above, and g,h E(S). Now, in 2 (T ), X 1 and Y( 2 ) ae adjacent, as ae X 2 and Y 1. Hence we can constuct cycles of lengths C v +j = +j, fo2 j s, by identifying the ends of each of the 2 above paths appopiately with ( ) those of the path C v X 1 X 2. Let C be a cycle of length + s so constucted. By ou constuction, C contains 2 edges fom the subgaph of 2 (T ) induced by W. Also, evey vetex of the complete gaph 2 (S) = K ( s is adjacent to all vetices of W. It follows that C can be extended to 2) ( ) ( ) ( ) cycles of each length fom s +s+1 to s +s ( ) It emains to show the existence of a cycle of length + 1. Fo this, choose thee 2 adjacent vetices, say, a 1 b 1, a 2 b 2, and a 3 b 3 on C v. Assume that a 1 is adjacent to a, and that a 3 is adjacent to b, fo some a,b E(T v ) (not necessaily distinct). Choose two adjacent edges e 1 and e 2 in S and eplace the P 3 fomed by the above thee vetices by the P 4 fomed by a 1 b 1, ae 1, be 2, a 3 b 3 to poduce the equied cycle. This completes the poof. Now this esult can easily be extended to all connected gaphs. Theoem If G is connected, with q 2, then 2 (G) is pancyclic. Poof. The poof is by induction on the numbe of cycles in G. The base case is coveed by the pevious theoem. Let e = uv be a cycle edge, and w a new vetex not in V(G). Constuct a gaph H = G e +f, whee f = uw. Then 2 (H) is isomophic to a spanning subgaph of 2 (G). ThenH is connected and has fewe cycles than G. Hence by the induction hypothesis, 2 (H) is pancyclic. It follows that 2 (G) is pancyclic. As we stated befoe, howeve, a much stonge esult than the one given in the above theoem holds. We state this in ou next theoem [2]. Theoem If G is a gaph with no isolated edges, then 2 (G) is vetex pancyclic. While it may be possible to extend this theoem to some gaphs having isolated edges, it cannot be done fo all gaphs, even to thei nontivial components. Fo example, let G be the disjoint union P 4 2K 2, then the nontivial component of 2 (G) is the complete multipatite gaph K 1,1,2,5, which clealy is not Hamiltonian, and thus is not pancyclic. Howeve, we believe that, fo a gaph G with at most one isolated edge, 2 (G) is pancyclic, and that it may even be vetex pancyclic. Futhe poblems along these lines suggest themselves. Fo example, find conditions on G unde which 2 (G) has isolated vetices but the nontivial component is Hamiltonian, pancyclic, o vetex-pancyclic. Anothe open poblem is to study 2 (G) in elation to Hamiltonian connectedness and panconnectedness Independence numbe. Let M be a set of independent edges in a gaph G. IfA and B ae -sets of M, thena and B ae nonadjacent in (G). Howeve, not all pais of nonadjacent vetices aise in this way. It is also the case that two -sets of edges of G ae nonadjacent in (G) if they geneate vetex-disjoint subgaphs. What we

8 1516 JAY BAGGA show is that when one consides a set of independent vetices in (G) of maximum ode, then with a few exceptional families of gaphs, it is poduced by a maximum independent set of edges of G. Weuseα(G) and α (G) fo the independence numbe and edge-independence ) numbe of G, espectively. We also denote the set of all -sets of a set X by. ( X Ou next esult [2] gives the independence numbe of (G) in tems of the edgeindependence numbe of G. Let G be a gaph with at least edges. Then the independence num- α ( (G) ) ( ) α (G) =. (3.1) Theoem be of (G) is Futhemoe, ) if S is a maximum independent set of vetices in (G), then eithe (i) S = fo some maximum independent set X of edges of G, o ( X (ii) S consists of +1 disjoint stas K 1,,o (iii) = 3 and the vetices in S ae K 1,3 s o K 3 s. Poof. If X is a maximum independent set of edges of G, then, clealy, -sets of X ae independent vetices in (G). Thus, α ( (G) ) ( ) α (G). (3.2) To pove the evese inequality, let V 1,V 2,...,V k be -sets of E(G) which ae independent vetices in (G). Also, let m = numbe of these sets which ae matchings in G, l = numbe of these sets which ae not matchings in) G, and h = numbe of edges of G in the union of the m matchings. Clealy, m.letu = V 1 V 2 V k.we obseve that if two edges of U ae adjacent in G, they must belong to the same V i and each such pai is in only one V i. Thus, we fom an independent set of edges in G by taking the h edges mentioned above, and one of the nonindependent edges of each of the l nonmatchings. Hence, l +h α (G) = α, and theefoe, ( h ( ) ( ) ( ) h l +h α k = l +m l +, (3.3) fom which we have the desied inequality. To pove the ( second ) pat, let S ={V 1,V 2,...,V k } be a maximum independent set in (G) with k = α. It follows fom (3.3) that ( ) ( ) ( ) h l +h α k = l +m = l + = =. (3.4) ( ) ( ) If l = 0, then k = m = h = α so that S satisfies (i). If = 1, then again all V i s ae) matchings ) so that l = 0. Thus we assume that l>0 and >1. In this case, l + = implies that h = 0andl = + 1. Consequently, α = + 1. Moeove, it ( h ( l+h

9 OLD AND NEW GENERALIZATIONS OF LINE GRAPHS 1517 follows that m = 0, so that no V i is a matching. If any V i is not a sta, then it has two independent edges o it is a K 3. In the fist case, one obtains + 2 independent edges in G, a contadiction. In the second case, it follows that = 3 so that each V i is a K 3 o a K 1,3. The above theoem chaacteizes the maximum independent sets in (G). Weobseve that moe can be said about the stuctue of G in cases (ii) and (iii), namely, that each additional edge in G must join the cente of one sta to some vetex in anothe component Othe popeties and types of supe line gaphs. We have descibed seveal diffeent stuctual and othe popeties of supe line gaphs. The wealth of esults indicates that this is a ich and fetile aea fo eseach. Some open poblems wee listed in the above subsections. Bagga et al. [1, 2, 3, 5, 6] have also studied vaiations such as supe line multigaphs and supe line digaphs. Howeve, only some peliminay wok has been done fo these vaiations and futhe exploations ae equied. 4. Tiangle gaphs. In this section, we descibe anothe ecent genealization of line gaphs. As noted in the intoduction, the vetices of the line gaph can be consideed as cliques of ode 2, with two being adjacent if they have a K 1 in common. This concept has been genealized to clique gaphs. We will mention the geneal case in the next section. Hee, we conside the special case of tiangles. The tiangle gaph of G, denoted by (G), isthegaphwithvetexsetthesetof tiangles in G. Two vetices ae adjacent in (G) if, as tiangles of G, they shae an edge in common. If G has no tiangle, then (G) is undefined. A gaph H is called a tiangle gaph if H (G) fo some G. Othewise it is called a nontiangle gaph. In [21], the poblem of detemining necessay and sufficient conditions fo a gaph to be a tiangle gaph was aised. In [7, 8], thee is some ecent pogess towads a solution to this poblem. We descibe some of these esults below Some classes of tiangle gaphs. We begin by listing seveal classes of gaphs which ae tiangle gaphs. It is easily seen that K n is a tiangle gaph since K n = (K 1,1,n ). Similaly, cycles and paths can be shown to be tiangle gaphs since C n (W n ),fon 4, whee W n is the wheel, and P n (W n e), whee e is a im edge. Some poducts. K m K n = (K 1,m,n ), K m C 3 = K m K 3 (K 1,m,3 ), K m C n = (K m C n ),(m,n 3), and K m P n = (K m P n+1 ) ae all tiangle gaphs. The Platonic gaphs. As Figue 4.1 shows, Q 3 = (octahedon). Also, tetahedon = K 4 = (K 1,1,4 ) and dodecahedon = (icosahedon). We will see below that the octahedon and the icosahedon ae nontiangle gaphs. Theoem 4.1. A tee H is a tiangle gaph if and only if (H) 3. Poof. Obseve that a gaph that contains K 1,4 as an induced subgaph is a nontiangle gaph. Convesely, if (H) 3, use induction on (V (H)). We next descibe a necessay condition involving K 4 e [8].

10 1518 JAY BAGGA Figue Figue 4.2 Theoem 4.2. If H is a tiangle gaph with K 4 e as an induced subgaph, then thee exists a vetex x in H such that x is adjacent to thee vetices of one tiangle of K 4 e and nonadjacent to the fouth vetex. Coollay 4.3. Coollay 4.4. Fo any n 4, K n e is not a tiangle gaph. The octahedon and the icosahedon ae nontiangle gaphs Tiangle labeling of a gaph. We next conside anothe class of gaphs that stictly includes the class of tiangle gaphs. A tiangle labeling of a gaph is defined to be a mapping f : V(H) N 3 (tiples of positive integes) such that xy E(H) if and only if f(x) f(y) =2. Figue 4.2 shows a tiangle labeling of the Petesen gaph. It can be shown by a diect agument that the Petesen gaph is not a tiangle gaph. Theoem 4.5. A tiangle gaph admits a tiangle labeling. The convese is not tue. This leads us to define some new classes of gaphs as follows. Let = the set of line gaphs, s = the set of induced subgaphs of line gaphs, = the set of tiangle gaphs,

11 OLD AND NEW GENERALIZATIONS OF LINE GRAPHS 1519 s = the set of induced subgaphs of tiangle gaphs, and l = the set of gaphs that admit a tiangle labeling. We then have the following esult [8]. Theoem 4.6. (i) = s. (ii) s. (iii) s = l. Fo some moe necessay conditions on tiangle gaphs, and some othe classes of tiangle gaphs, we efe the eade to [8]. The geneal chaacteization of tiangle gaphs is an open poblem and futhe exploations in this aea should be undetaken Open poblems. We conclude ou discussion of tiangle gaphs by listing a few poblems and diections fo moe investigations in this aea. Of couse, it would be nice to have a chaacteization of tiangle gaphs. As mentioned above, [8] has made some pogess in this diection by obtaining a numbe of necessay conditions, including a fobidden subgaph condition. We saw ealie that seveal poduct gaphs ae tiangle gaphs. Simila questions can also be asked fo othe poducts; in paticula, fo which gaphs G is the Catesian poduct K n G a tiangle gaph? We obseve that (K 5 ) = L(K 5 ). One wishes to chaacteize G fo which (G) L(G). Similaly, when is (G) G? LetG 1 and G 2 be gaphs in which each edge belongs to a tiangle. Unde what conditions does (G 1 ) (G 2 ) imply that G 1 G 2? Motivated by the definition of line completion numbe defined in Section 3.2, we can define the tiangle completion numbe, tc(g), to be the minimum numbe of edges to be added to G so that the esulting gaph becomes a tiangle gaph. As a fist step, one wants bounds fo tc(g) fo a given gaph G. 5. Some moe genealizations. It was mentioned at the beginning of Section 4 that tiangle gaphs ae a special case of clique gaphs, a moe geneal vaiation on line gaphs. In this section, we biefly descibe clique gaphs and some othe vaiations. Fo moe details and esults, we efe the eade to Pisne [22]. We obseve that by a clique of a gaph G, we mean a complete subgaph of G. Some authos (including Pisne [22]) use the tem simplex fo a complete subgaph and clique fo inclusion-maximal simplices. Since an edge is a clique of ode 2, a genealization of the line gaph L(G) of agaphg is obtained when one consides all cliques of G of a fixed ode. Depending on how one defines adjacency, seveal vaiations ae possible, as we see below. The following definitions ae fom [22]. (i) The k-gallai gaph k (G) has all K k s in G as vetices, with two adjacent if thei union induces a K k+1 e. Obseve that 1 (G) = G. (ii) The k-in-m gaph Φ k,m has all K k s in G as vetices, with two adjacent if they lie in a common K fo some m. (iii) The k-line gaph L k (k 2) has all K k s in G as vetices, with two adjacent if thei intesection is a K k 1. We obseve that L k (G) is the edge-disjoint union of Φ k,k+1 (G) and k (G). (iv) The k-ovelap clique gaph C k (G) of G has all maximal cliques of G as vetices, with two adjacent wheneve thei intesection contains at most k vetices.

12 1520 JAY BAGGA (v) The cycle gaph Cy(G) of a gaph G has all induced cycles of G as vetices, with two adjacent if they shae some common edge. Fo moe such genealizations and thei popeties, we efe the eade to [22]. 6. Conclusion. In this bief suvey, we have descibed line gaphs and thei genealizations. Many of the genealizations ae well known and well studied. Othes ae moe ecent. Fo all genealizations, a numbe of poblems and aeas of futhe study have been pesented. This in an active aea of eseach. We have included a set of efeences which have been cited in ou desciption. These efeences ae just a small pat of the liteatue, but they should povide a good stat fo eades inteested in this aea. Acknowledgment. The autho wishes to expess his gatitude to all of his coauthos listed in the efeences. The autho also thanks the efeees fo thei helpful comments. Refeences [1] J.S.Bagga,L.W.Beineke,andB.N.Vama,Supe line gaphs and thei popeties, Combinatoics, Gaph Theoy, Algoithms and Applications (Beijing, 1993), Wold Scientific Publishing, New Jesey, 1994, pp [2], Independence and cycles in supe line gaphs, Austalas. J. Combin. 19 (1999), [3], The supe line gaph 2, Discete Math. 206 (1999), no. 1 3, [4] K. J. Bagga and M. R. Vasquez, The supe line gaph 2 fo hypecubes, Cong.Nume.93 (1993), [5] K.S.Bagga,L.W.Beineke,andB.N.Vama,The line completion numbe of a gaph, Gaph Theoy, Combinatoics, and Algoithms, Vol. 1, 2 (Kalamazoo, Mich, 1992), Wiley- Intescience, New Yok, 1995, pp [6], Supe line gaphs, Gaph Theoy, Combinatoics, and Algoithms, Vol. 1, 2 (Kalamazoo, Mich, 1992), Wiley-Intescience, New Yok, 1995, pp [7] R. Balakishnan, Tiangle gaphs, Gaph Connections (Cochin, 1998), Allied Publishes, New Delhi, 1999, p. 44. [8] R. Balakishnan, J. Bagga, R. Sampathkuma, and N. Thillaigovindan, Tiangle gaphs, pepint, [9] R. Balakishnan and K. Ranganathan, A Textbook of Gaph Theoy, Univesitext, Spinge- Velag, New Yok, [10] M. Behzad and G. Chatand, Total gaphs and tavesability, Poc. Edinbugh Math. Soc. (2) 15 (1966/1967), [11] L. W. Beineke, Chaacteizations of deived gaphs, J. Combinatoial Theoy 9 (1970), [12] H. J. Boesma and C. Hoede, Path gaphs, J. Gaph Theoy 13 (1989), no. 4, [13] G. Chatand, On hamiltonian line-gaphs, Tans. Ame. Math. Soc. 134 (1968), [14] G. Chatand, H. Hevia, E. B. Jaett, F. Saba, and D. W. VandeJagt, Subgaph distance and genealized line gaphs, Gaph Theoy, Combinatoics, Algoithms, and Applications (San Fancisco, Calif, 1989), SIAM, Philadelphia, 1991, pp [15] A. G. Chetwynd, Total colouings of gaphs a pogess epot, Gaph Theoy, Combinatoics, and Applications, Vol. 1 (Kalamazoo, Mich, 1988), Wiley-Intescience, New Yok, 1991, pp [16] F. Haay, Gaph Theoy, Addison-Wesley, Massachusetts, [17] F. Haay and R. Z. Noman, Some popeties of line digaphs, Rend. Cic. Mat. Palemo (2) 9 (1960),

13 OLD AND NEW GENERALIZATIONS OF LINE GRAPHS 1521 [18] R. L. Hemminge and L. W. Beineke, Line gaphs and line digaphs, Selected Topics in Gaph Theoy (W. B. Lowell and R. J. Wilson, eds.), Academic Pess, New Yok, 1978, pp [19] J. Kausz, Démonstation nouvelle d une théoème de Whitney su les éseaux, Mat. Fiz. Lapok 50 (1943), (Hungaian). [20] J. Mitchem, Hamiltonian and Euleian popeties of entie gaphs, Gaph Theoy and Applications (Poc. Conf., Westen Michigan Univ., Kalamazoo, Mich, 1972; Dedicated to the Memoy of J. W. T. Youngs), Lectue Notes in Math., vol. 303, Spinge, Belin, 1972, pp [21] S. D. Monson, N. J. Pullman, and R. Rees, A suvey of clique and biclique coveings and factoizations of (0, 1)-matices, Bull. Inst. Combin. Appl. 14 (1995), [22] E. Pisne, Gaph Dynamics, Pitman Reseach Notes in Mathematics Seies, vol. 338, Longman, Halow, [23] A. C. M. van Rooij and H. S. Wilf, The intechange gaph of a finite gaph, Acta Math. Acad. Sci. Hunga. 16 (1965), [24] H. Whitney, Conguent gaphs and the connectivity of gaphs, Ame. J. Math. 54 (1932), Jay Bagga: Depatment of Compute Science, Ball State Univesity, Muncie, IN 47306, USA addess: jbagga@bsu.edu

14 Mathematical Poblems in Engineeing Special Issue on Modeling Expeimental Nonlinea Dynamics and Chaotic Scenaios Call fo Papes Thinking about nonlineaity in engineeing aeas, up to the 70s, was focused on intentionally built nonlinea pats in ode to impove the opeational chaacteistics of a device o system. Keying, satuation, hysteetic phenomena, and dead zones wee added to existing devices inceasing thei behavio divesity and pecision. In this context, an intinsic nonlineaity was teated just as a linea appoximation, aound equilibium points. Inspied on the ediscoveing of the ichness of nonlinea and chaotic phenomena, enginees stated using analytical tools fom Qualitative Theoy of Diffeential Equations, allowing moe pecise analysis and synthesis, in ode to poduce new vital poducts and sevices. Bifucation theoy, dynamical systems and chaos stated to be pat of the mandatoy set of tools fo design enginees. This poposed special edition of the Mathematical Poblems in Engineeing aims to povide a pictue of the impotance of the bifucation theoy, elating it with nonlinea and chaotic dynamics fo natual and engineeed systems. Ideas of how this dynamics can be captued though pecisely tailoed eal and numeical expeiments and undestanding by the combination of specific tools that associate dynamical system theoy and geometic tools in a vey cleve, sophisticated, and at the same time simple and unique analytical envionment ae the subject of this issue, allowing new methods to design high-pecision devices and equipment. Authos should follow the Mathematical Poblems in Engineeing manuscipt fomat descibed at Pospective authos should submit an electonic copy of thei complete manuscipt though the jounal Manuscipt Tacking System at mts.hindawi.com/ accoding to the following timetable: Guest Editos José Robeto Castilho Piqueia, Telecommunication and Contol Engineeing Depatment, Polytechnic School, The Univesity of São Paulo, São Paulo, Bazil; piqueia@lac.usp.b Elbet E. Nehe Macau, Laboatóio Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, São Paulo, Bazil ; elbet@lac.inpe.b Celso Gebogi, Cente fo Applied Dynamics Reseach, King s College, Univesity of Abedeen, Abedeen AB24 3UE, UK; gebogi@abdn.ac.uk Manuscipt Due Decembe 1, 2008 Fist Round of Reviews Mach 1, 2009 Publication Date June 1, 2009 Hindawi Publishing Copoation