A Survey of Solved Problems and Applications on Bandwidth, Edgesum and Pro le of Graphs Yung-Ling Lai National Chiayi Teacher College Chiayi, Taiwan, R.O.C. Kenneth Williams Western Michigan University Abstract This pap er provides a survey of results on the exact bandwidth, edge- sum, and pro le of graphs. A bibliographyof work in these areas is pro- vided. The emphasis is on comp osite graphs. This may be regarded as an up date of the original survey of solved bandwidth problems by Chinn, Chvatalova, Dewdney, and Gibbs[10] in 1982. Also several of the applica- tion areas involving these graph parameters are describ ed. 1 Intro duction and terminology For a graph G, V (G) denotes the set of vertices of G and E (G) denotes the set of edges of G. Let G = (V; E ) be a graph on n vertices. A 1-1 mapping f : V ! f1; 2;:::;ng is called a proper numbering of G. The bandwidth B (G) of aproper f numbering f of G is the number B (G)= maxfjf (u) f (v )j : uv 2 E g; f and the bandwidth B(G) of G is the number B (G)= minfB (G): f is a prop er numb ering of Gg: f A prop er numb ering f is called a bandwidth numbering of G if B (G)=B (G). f For example, Figure 1 shows bandwidth numb erings for the graphs P ;C ;K 4 5 1;4 and K . In general, B (P ) = 1, B (C ) = 2, B (K ) = dn=2e and B (K ) = 2;3 n n 1;n m;n m + dn=2e1form n. 1 t t t t 1 1 1 2 3 4 , l J t t l , , J l L L l t 2 l , , l J L L , t t t l 2 3 3 , l J L L J t 3 , l J L L , J , L L l t t t J J 1 2 5 t t t t t J J 4 4 5 4 5 Figure 1: Bandwidth numb erings for P ;C ;K and K . 4 5 1;4 2;3 For a prop er numb ering f ,the edgesum s (G) produced by f is given by f X s (G)= jf (u) f (v )j. f uv 2E (G) The edgesum s(G) of a graph G is de ned by s(G)=minfs (G):f is a prop er numb ering of Gg: f A prop er numb ering that achieves the edgesum of a graph is called an edge- sum numbering. The term here called edgesum means exactly the same as the terms bandwidth sum or minimum sum which have b een used by some previous investigators. For example, Figure 2 shows edgesum numb erings for the graphs P ;C ;K 4 5 1;4 and K . In general, s(P ) = n 1 and s(C )=2(n 1). Yao and Wang[100] 2;3 n n showed that 8 n(n +2) > > > if n 0 (mo d 2) > < 4 s(K )= 1;n > > > n +1 > 2 : ( ) if n 1 (mo d 2) 2 and, for m n, Williams[96] established that 8 2 3 2 > 3n m m +6m n +4m > > if n m 0( mo d 2) > > < 12 s(K )= m;n > > 2 3 2 > 3n m m +6m n + m > > : if n m 1( mo d 2): 12 Also, for a prop er numb ering f , the pro le width w (v ) of a vertex v in a f graph G is the number w (v )= max (f (v ) f (x)) f x2N [v ] 2 t t t t 1 1 1 2 2 4 , l J t t l , , J l L L l t 2 l , , l J L L , t t t 2 5 l 3 , l J L L J t 3 , l J L L , J , L L l t t t J J 1 3 5 t t t t t J J 4 4 5 3 4 Figure 2: Edgesum numb erings for P ;C ;K and K . 4 5 1;4 2;3 t t t t 1 1 1 2 4 5 , l J t t l , , J L L l l t 2 l , , l J L L , t t t l 2 5 5 , l J L L J t 4 , l J L L , J , L L t t t l J J 1 2 3 t t t t J 3 t J 3 4 3 4 Figure 3: Pro le numb erings for P ;C ;K and K . 4 5 1;4 2;3 where N [v ]=fx 2 V : x = v or xv 2 E g is the closed neighborhood of v . Since v 2 N [v ], we have w (v ) = 0 if f (v ) f (x) for all x 2 N [v ]. The pro le (also f called skyline) P (G) of a proper numbering f of G is de ned by f X P (G)= w (v ) f f v 2V and the pro le P(G) of G is the number P (G) = minfP (G): f is a prop er numb ering of Gg: f A prop er numb ering f is called a pro le numbering of G if P (G)= P (G). f For example, Figure 3 shows pro le numb erings for the graphs P , C , K 4 5 1;4 and K . In general, P (P ) = n 1, P (C ) = 2n 3, P (K ) = n and, for 2;3 n n 1;n m n, P (K )=mn + m(m 1)=2 (see Lin and Yuan[65]). m;n For a prop er numb ering f on a graph G of order n,the complementary num- 0 0 bering f on G is de ned by f (v )= n +1 f (v ) for each vertex v in G. Then 0 0 B (G)= B (G) and s (G)=s (G). Thus the complementary numb ering of any f f f f bandwidth (or edgesum) numb ering is also a bandwidth (or edgesum) number- ing. However this relation do es not hold for pro le numb erings, as is illustrated in Figure 4. The decision problem corresp onding to nding the bandwidth of an arbi- trary graph was shown to be NP-complete by Papadimitriou[82]. Garey, Gra- ham, Johnson and Knuth[25] showed that the problem is NP-complete even for 3 1 4 v v v v 4 1 @ @ @ @ 2 3 3 2 v v v v @ @ 0 P (G)=3 P (G)=6 f f Figure 4: Complementary numb erings, di erent pro le results. trees of maximum degree 3. The decision problem asso ciated with determin- ing the edgesum for an arbitrary graph (sometimes called linear layout or the linear arrangement problem) was shown to be NP-complete by Garey, Johnson and Sto ckmeyer[27 ]; but the edgesum for trees is p olynomal. In fact, Chung[14 ] provided the most ecient algorithm known to date to achieve an edgesum num- b ering for arbitrary trees. Lin and Yuan[65] indicated that the pro le minimiza- tion problem of an arbitrary graph is equivalenttotheinterval graph completion problem, which was shown to be NP-complete by Garey and Johnson[26]. Kuo and Chang[52] provide a p olynomial algorithm to achieve a pro le numb ering for an arbitrary tree of order n. Wesay that the bandwidth, edgesum, or pro le numb ering problem is solved for a class of graphs if a p olynomial time algorithm is known to achieve the numb ering, or if a formula to describ e the parameter value (bandwidth, edgesum or pro le) is known. 2 Survey of results The reader should esp ecially note the original survey pap er on bandwidth by Chinn, J. Chvatalova, Dewdney, and Gibbs [10]. This provides many of the key concepts and inequalities up on which more recentwork is based. Some additional survey material is included in Chung[13]. Another excellent resource is Chapter 21, by Miller, of Applications of Discrete Mathematics [76]. A large number of approximation algorithms for the bandwidth and pro le of a variety of graphs have b een given. Approximation algorithms for general graphs include those given in Cheng[9 ], Cuthill and McKee[19 ], Gibbs, Poole, and Sto ckmeyer[28 ], GowriSankaran and Opatrny[31 ], Je [44], King[48], Ko o and Lee[50], Lai, Weingarten and Eshraghi[53], Luo[72], Quo c and O'Leary[83], Snay[88 ], and Wiegers and Monien[95]. Everstine[23 ], Hein and Groten[40] and Smyth[87] provide comparisons of some of these algorithms. Approximation al- 4 gorithms for trees or caterpillars are given in GowriSankaran, Miller and Opatrny [30], Odlyzko and Wilf[80], and Haralambides, Makedon and Monien[33]. A considerable amount of work has b een published providing bounds on the bandwidth of graphs. Harp er (see [38] and [39]) rst worked on bandwidth and edgesum problems and he develop ed some of the techniques and approaches to problems that have since b ecome widely used. The basic lower bound for band- width was established in [38]. It has since b een re ned by Hare and Hare[35] and by Li and Lin[63]. Anumb er of the imp ortant b ounds on bandwidth are provided in the survey pap er [10]. We will now list some of the more recent work. Let D (G) denote the diameter of connected graph G.