Perhaps the method will give that for every e > U f() > p - 3/+e There is o o-trivial upper boud for f() ad ot eve f() < Z - e. seems to be kow, where

Transcription

1 ON MAXIMUM CHORDAL SUBGRAPH * Paul Erdos Mathematical Istitute of the Hugaria Academy of Scieces ad Reu Laskar Clemso Uiversity 1. Let G() deote a udirected graph, with vertices ad V(G) deote the vertex set, E(G) deote the edge set. Let e(g) deote the umber of edges of G. It wí11 be assumed that G() has o loops or multiple edges. A graph G() is chordal (triagulated, rigid circuit) if every cycle of legth > 3 has a chord : amely a edge joiig two ocosecutive vertices o the cycle. The class of chordal graph iclude trees, k-trees, complete graphs ad iterval graphs. Chordal graphs have applicatio i facility locatio [], schedulig problems [8], ad i the solutio of sparse systems of liear equatio 110]. Such graphs are also kow to be perfect. Certai problems that are kow to be NP-hard for geeral graphs ca be solved i polyomial time for chordal graphs [6]. As a result, chordal graphs have bee studied by may, e.g. [1], (5]. If a graph is ot chordal, the followig questios are quite appropriate to ask : 1) What is the miimal set of ew edges to be added to the graph to make it chordal? ) What is the miimal set of edges to be deleted from the graph such that the resultig graph is a maximum chordal subgraph? Rose, Tarja ad Lueker have aswered 1) algorithmically [9]. I aswer to ) recetly Dearig, Shier, ad Warer [3] have developed a polyomial algorithm to geerate a maximal chordal subgraph. It may be poited out that their algorithm does ot geerate a maximum chordal subgraph. I aswer to 1) Erdos [4] showed that for some positive e > 0 f() > z - -c t This paper is i fial form ad will ot appear elsewhere. CONGRESSUS NUMERANTIUM, VOL. 39 (1983), pp

2 Perhaps the method will give that for every e > U f() > p - 3/+e There is o o-trivial upper boud for f() ad ot eve f() < Z - e. seems to be kow, where f() is the smallest iteger so that for every graph with vertices oe ca add <_ f() ew edges so that the resultig graph would be chordal. This ote determies asymptotically the miimum umber of edges to be deleted from the graph such that the resultig graph is a maximum chordal subgraph.. Deote by f() the smallest iteger such that every graph G() with vertices ca be made chordal be deletig <- f() edges of G(). Theorem 1. f() <- - ( 1+o(l»V 31 ~, where o(1) ' o as 1 Proof : Let T(,t) be the Tura graph [111, i.e. a complete t-partite graph with vertices with approximately vertices i each color t class. Let v l, y,.., v t deote the color classes. The umber of edges of T(,t) e(t,(,t)) CZ) - t Z (1-1) (1) Let G 1 be a spaig chordal subgraph of T(,t) havig maximum umber of edges. Clearly G 1 caot cotai ay cycle whose vertices are i t o differet color classes. Thus, the iduced subgraph of ay two color classes of G 1 must be a tree. Hece, e(g 1 ) ( t)( - 1) (-') 368

3 Let H be a spaig subgraph of T(,t) descríbrd as 1)elow : Cosider xí e l' í,. joi each k t : {x 1' x '.. ' x t ), xx to all vertices of V., i 4 j, j=1,,.., Clearly H is chordal beig a split graph 17). Also e(h) - ( Z)( - 1). Thus by () H is a spaig chordal subgraph of T(,t) of maximum size. The umber of edges to be deleted from T(,t) to obtai H is (1-t) - ()( t 1) t _ t++ t t Now take t =. The (3) becomes - ( 1+o(1))V ~~` where o(1) - 0 as ~ -. Thus, f(), - ( l+o(1)i 3/ Theorem. / f() < - (1 - e) 3 for every e > 0 if > 0 (e). Proof : We ca assume that G() has at least - (1 - e)d_^ 3/ edges. Suppose G() has () - e l edges ad that the largest chordal subgraph of G() has e edges. It suffices to prove e 3/ (1 - e) (4) Let rj > 0 be small, much smaller tha e. he costruct a subgraph G(m) cosistig of m vertices each of which has degree > (1-q). The co structío is as follows : delete from G() a vertex x l (if ay) of degree 5 (1-) ad cotiue this process as log as possible. Suppose after k such steps we obtai G(m) = G(x l`+1, x K+,.., x ) each vertex of which has degree > (1-). Claim k c C (5) t. 369

4 s., here C E depeds o E ad ad ot o. To prove the claim observe that if k = [C J] e(g()) < ( m) + k(l (-k) + k(1 - f) (6) + k - rlk 3/ ( - if CE is sufficietly large. Thus by (4) we have othig to prove, Cosider G(m). Assume that t is the largest positive iteger such that G(m) cotais a K t. The by Tura's theorem G(m) has at most M (1 - t) edges (7) Let Kt = (y l, y ' " ' yt). Now deg G(m) y i > (1-), for each i, i=1,,.., t. Adjoi all the edges joiig to each y to vertices outside of i K t. This produces a chordal subgraph with at least t((l-) - (t-1)) edges t(1-) - (8) Now if t = [4v], the (8) becomes 411 3/ (l-) - 16 which is 3/ / Thus, we get a chordal subgraph k,hose umber of edges > (9) This proves (4). Thus t < 4ti '1Q1 370

5 Now from (7) ad (8) ` el + e > mt t(1-) t > t t(1-) 16 (-k) t + t(1-) - 16 t t + t + t(1-q) 16 3/ C e, t + t(1-) - 16 (1-E),/- 3/ sice to + t is miimum if t = ad other terms ca be absorbed ito e 3/ Thus combiig theorems 1 ad we ca state Theorem 3. f() = - (1 + 0(1)) 3/. Perhaps the followig problem is of some iterest ad deserves some study. Let f( ;t) be the smallest iteger for which every G( ;f( ;t)) cotais a chordal subgraph of t edges. At the momet we oly kow that f(,) _ [4] + 1 (11) We do ot give the details of the proof of (11) but oly idicate some of the ecessary steps. We hope to retur to this problem i the future. Observe that the complete bipartite graph of white ad +1 black vertices immediately shot: that f(,)? 4. To prove the upper boud i (11) we oly remark that this immediately follows if our G(,[, ]+1) is assumed to be coected. For the by Tura's theorem our graph cotais a triagle ad by a simple argumet this triagle ca be exteded to a chordal graph of edges. If the graph is ot coected somewhat more 37 1

6 complicated methods are eeded, which as we stated we hope to discuss at aother occasio. Just oe more remark. It is well kow that every z graph of G( ;[ 4 ]+1) cotais a edge (x,,x ) ad o further vertices x i joied to both x l ad x. This implies i fact that f(,(l + t)) _ [4 ] + 1 for sufficietly small E > 0. Ackowledgemet The secod author thakfully ackowledges the partial support from the Natioal Sciece Foudatio Grat #ISP (EPSCoR) to do this research. 37

7 Refereces 1. P. Buema, "A characterizatio of rigid circuit graphs." Discrete "fathematícs, 9 (1974) R. Chadrasekara ad A. Tamir, "Polymoially bouded algorithms for locatig p-ceters o a tree." Discussio paper No. 358, Ceter for Mathematical Studies i Ecoomics ad Maagemet Sciece, -Northwester Uiversity, (1978). 3. P. M. Dearig, D. R. Shier, D. D. Warer, "Maximal Chordal Subgraphs." Clemso Uiversity Techical) Report ;'404, December P. Erdos, "Some ew applicatios of probability methods to combiatorial aalysis ad graph theory." Proc. 5th S. E. Cof. Combiatorics, Graph Theory, ad Computig, D. R. Fulkerso ad 0. A. Gross, "Icidece matrices ad iterval graphs." Pacific J. Math, 15 (1965) F. Gavril, "Algorithms for miimum collorig, maximum clique, miimum coverig by cliques, ad maximum idepedet set of chordal graph," SIAM J. Comput., 1 (197) M. Golumbic, "Algorithmic Graph Theory ad Perfect Graphs." Academic Press New York, (1980). 8 C. Papedimitrio ad M. Yaakakis, "Schedulig iterval-ordered tasks." SIAM J. Comput., 8 (1979) D. Rose, R. Tarja ad G. Lueker, "Algorithmic aspects of vertex elimiatio o graphs." SIAM J. Comput., 5 (1976) D. Rose, "A graph-theoretic study of the umerical solutio of sparse positive defiite systems of liear equatios." (R. Read, ed.), Graph Theory ad Computig, Academic Press, New York, (197) P. Tura, "O a extremal problem i graph theory." (I Hugaria), Mat. Fiz. Lapok, 48 (1941)