Tree Spanners on Chordal Graphs: Complexity, Algorithms, Open Problems

Transcription

1 Tree Spanners on Chordal Graphs: Complexty, Algorthms, Open Problems A. Brandstädt 1, F.F. Dragan 2, H.-O. Le 1, and V.B. Le 1 1 Insttut für Theoretsche Informatk, Fachberech Informatk, Unverstät Rostock, Rostock, Germany. {ab,hoang-oanh.le,le}@nformatk.un-rostock.de 2 Dept.of Computer Scence, Kent State Unversty, Oho, USA. dragan@cs.kent.edu Abstract. A tree t-spanner T n a graph G s a spannng tree of G such that the dstance n T between every par of vertces s at most t tmes ther dstance n G.The Tree t-spanner problem asks whether a graph admts a tree t-spanner, gven t.we substantally strengthen the hardness result of Ca and Cornel [SIAM J. Dscrete Math. 8 (1995) ] by showng that, for any t 4, Tree t-spanner s NP-complete even on chordal graphs of dameter at most t+1 (f t s even), respectvely, at most t + 2 (f t s odd).then we pont out that every chordal graph of dameter at most t 1 (respectvely, t 2) admts a tree t-spanner whenever t 2 s even (respectvely, t 3 s odd), and such a tree spanner can be constructed n lnear tme. The complexty status of Tree 3-Spanner stll remans open for chordal graphs, even on the subclass of undrected path graphs that are strongly chordal as well.for other mportant subclasses of chordal graphs, such as very strongly chordal graphs (contanng all nterval graphs), 1-splt graphs (contanng all splt graphs) and chordal graphs of dameter at most 2, we are able to decde Tree 3-Spanner effcently. 1 Introducton and Results All graphs consdered are connected. For two vertces n a graph G, d G (x, y) denotes the dstance between x and y;that s, the number of edges n a shortest path n G jonng x and y. The value dam(g) :=maxd G (x, y) sthedameter of the graph G. Let t 2 be a fxed nteger. A spannng tree T of a graph G s a tree t- spanner of G f, for every par of vertces x, y of G, d T (x, y) t d G (x, y). Tree t-spanner s the followng problem: Gven a graph G, does G admt a tree t-spanner? There are many applcatons of tree spanners n dfferent areas;especally n dstrbuted systems and communcaton networks. In [1], for example, t was shown that tree spanners can be used as models for broadcast operatons;see also [21]. Moreover, tree spanners also appear n bology [2], and n [25], tree Research of ths author was supported by DFG, Project no.br1446-4/1 P. Bose and P. Morn (Eds.): ISAAC 2002, LNCS 2518, pp , c Sprnger-Verlag Berln Hedelberg 2002

2 164 A.Brandstädt et al. spanners were used n approxmatng the bandwdth of graphs. We refer to [7,6, 22,24] for more background nformaton on tree spanners. In [6] Ca and Cornel gave a lnear tme algorthm solvng Tree 2-Spanner and proved that Tree t-spanner s NP-complete for any t 4. A graph s chordal f t does not contan any chordless cycle of length at least four. For a popular subclass of chordal graph, the strongly chordal graphs, Brandstädt et al. [3] proved that, for every t 4, Tree t-spanner s solvable n lnear tme. Indeed, they show that every strongly chordal graph admts a tree 4-spanner. In contrast, one of our results s Theorem 1 For any t 4, Tree t-spanner s NP-complete on chordal graphs of dameter at most t +1 (f t s even), respectvely, of at most t +2 (f t s odd). Comparng wth a recent result due to Papoutsaks [20], t s nterestng to note that the unon of two tree t-spanners, t 4, may contan chordless cycles of any length. Ths perhaps ndcates the dffculty n provng Theorem 1. Indeed, our reducton from 3SAT to Tree t-spanner gven n Secton 2 s qute nvolved. Moreover, to the best of our knowledge, Theorem 1 s the frst hardness result for Tree t-spanner on a restrcted, well-understood graph class. Notce that n [12] t s shown that Tree t-spanner, t 4, s NP-complete on planar graphs f the nteger t s part of the nput. In vew of the dameter constrants n Theorem 1, we note that Tree t- Spanner remans open on chordal graphs of dameter t (t s even) and of dameter t 1, t or t + 1 (f t s odd). For smaller dameter we have Theorem 2 For any even nteger t, every chordal graph of dameter at most t 1 admts a tree t-spanner, and such a tree spanner can be constructed n lnear tme. For any odd nteger t, every chordal graph of dameter at most t 2 admts a tree t-spanner, and such a tree spanner can be constructed n lnear tme. We were able also to show that chordal graphs of dameter at most t 1 (t s odd) admt tree t-spanners f and only f chordal graphs of dameter 2 admt tree 3-spanners. Ths result s used to show that every chordal graph of dameter at most t 1(t s odd), f t s planar or a k-tree, for k 3, has a tree t-spanner and such a tree spanner can be constructed n polynomal tme. Note that, for any fxed t, there s a 2-tree wthout a tree t-spanner [16]. So, even those knd of results are of nterest. Unfortunately, the reducton above (from arbtrary odd t to t = 3) s of no drect use for general chordal graphs because not every chordal graph of dameter at most 2 admts a tree 3-spanner. One of our theorems characterzes those chordal graphs of dameter at most 2 that admt such spanners. We now dscuss Tree t-spanner on mportant subclasses of chordal graphs. It s well-known that chordal graphs are exactly the ntersecton graphs of subtrees n a tree [13]. Thus, ntersecton graphs of paths n a tree, called path graphs, form a natural subclass of chordal graphs. Tree t-spanner remans unresolved even on ths natural subclass of chordal graphs. The complexty status of Tree 3-Spanner remans a long standng open problem. However, t can be solved effcently for many partcular graph classes,

3 Tree Spanners on Chordal Graphs 165 such as cographs and complements of bpartte graphs [5], drected path graphs [17] (hence for all nterval graphs [16,18,23]), splt graphs [5,16,25], permutaton graphs and regular bpartte graphs [18], convex bpartte graphs [25], and recently for planar graphs [12]. In [5,19,20], some propertes of graphs admttng a tree 3-spanner are dscussed. On chordal graphs, however, Tree 3-Spanner remans even open on path graphs whch are strongly chordal as well. For some mportant subclasses of chordal graphs we can decde Tree 3-Spanner effcently. Graphs consdered n the theorem below are defned n Secton 5. Theorem 3 All very strongly chordal graphs and all 1-splt graphs admt a tree 3-spanner, and such a tree 3-spanner can be constructed n lnear tme. Theorem 4 For a gven chordal graph G =(V,E) of dameter at most 2, Tree 3-Spanner can be decded n O( V E ) tme. Moreover, a tree 3-spanner of G, f t exsts, can be constructed wthn the same tme bound. Theorem 3 mproves prevous results on tree 3-spanners n nterval graphs [16,18, 23] and on splt graphs [5,16,25]. The complexty status of Tree t-spanner on chordal graphs consdered n ths paper s summarzed n Fgure 1 and Table 1. chordal NP-c (t 4) t =3:open path t 3: open strongly chordal strongly path dameter 3 chordal 2-splt tree 4-spanner admssble t =3:open very strongly chordal ptolemac drected path nterval splt 1-splt tree 3-spanner admssble Fg. 1. The complexty status of Tree t-spanner on chordal graphs and mportant subclasses

4 166 A.Brandstädt et al. Table 1. The complexty status of Tree t-spanner on chordal graphs under dameter constrants Dameter at most Complexty t +2,t 5odd NP-complete t +1,t 4even NP-complete t +1,t 3odd? t, t 3? t 1, t 5odd? t 1, t =3 polynomal tme t 1, t 2even lnear tme t 2, t 3odd lnear tme Notatons and defntons not gven here may be found n any standard textbook on graphs and algorthms. We wrte xy for the edge jonng vertces x, y; x and y are also called endvertces of xy. For a set C of vertces, N(C) denotes the set of all vertces outsde C adjacent to a vertex n C; N(x) stands for N({x}) and deg(x) stands for N(x). We set d(v, C) :=mn{d(v, x) : x C}. The eccentrcty of a vertex v n G s the maxmum dstance from v to other vertces n G. The radus r(g) ofg s the mnmum of all eccentrctes and the dameter dam(g) ofg s the maxmum of all eccentrctes. A cutset of a graph s a set of vertces whose deleton dsconnects the graph. A graph s non-separable f t has no one-element cutset, and trconnected f t has no cutset wth 2 vertces. Blocks n a graph are maxmal non-separable subgraphs of that graph. Clearly, a graph contans a tree t-spanner f and only f each of ts blocks contans a tree t-spanner. Note also that dvdng a graph nto blocks can be done n lnear tme. Thus, n ths paper, we consder non-separable graphs only. Graphs havng a tree t-spanner are called tree t-spanner admssble. Fnally, we wll use of the followng fact n checkng whether a spannng tree s a tree t-spanner. Proposton 1 ([6]) A spannng tree T of a graph G satreet-spanner f and only f, for every edge xy of G, d T (x, y) t. 2NP-Completeness, t 4 In ths secton we wll show that, for any fxed t 4, Tree t-spanner s NPcomplete on chordal graphs. The proof s a reducton from 3SAT, for whch the followng famly of chordal graphs wll play an mportant role. Frst, S 1 [x, y] stands for a trangle wth two labeled vertces x and y. Next, for a fxed nteger k 1, S k+1 [x, y] s obtaned from S k [x, y] by takng to every edge e xy n S k [x, y] that belongs to exactly one trangle a new vertex v e and jonng v e to exactly the two endvertces of e. We wrte also S k for S k [x, y] for some sutable labeled vertces x, y. See Fgure 2.

5 Tree Spanners on Chordal Graphs 167 x y x y x y S 1 [x, y] S 2 [x, y] S 3 [x, y] a c x y b d S k 1 [a, b] S k 1 [c, d] S k [x, y] Fg. 2. The graph S k [x, y] obtaned from S k 1 [a, b] and S k 1 [c, d] by dentfyng b = d and jonng x = a wth y = c Equvalently, S k+1 [x, y] s obtaned from two dsjont S k [a, b] and S k [c, d] by dentfyng the two vertces c, d to a vertex z and jonng the vertces x := a and y := b by an edge. Wth ths notaton, z s the common neghbor of x and y n S k+1 [x, y], and we call S k [x, z] and S k [y, z] the two correspondng S k n S k+1 [x, y]. We denote by S k [x, y) the graph S k [x, y] y, that s, the graph obtaned from S k [x, y] by deletng the vertex y. The followng observatons collect basc facts on S k used n the reducton later. Observaton 1 (1) For every v S k [x, y], d Sk [x,y](v, {x, y}) k 2, (2) S k [x, y] hasatree(k +1)-spanner contanng the edge xy, (3) S k [x, y) hasatreek-spanner T such that, for each neghbor y of y n S k [x, y), d T (x, y ) k. Proofs of these and all other results are omtted n ths extended abstract. They wll be gven n the journal verson. Observaton 2 Let H be an arbtrary graph and let e be an arbtrary edge of H. LetK be an S k [x, y] dsjont from H. LetG be the graph obtaned from H and K by dentfyng the edges e and xy; see Fgure 3. Suppose that T satree t-spanner n G, t>k, such that the xy-path n T belongs to H. Then (1) d T (x, y) t k, and (2) there exsts an edge uv K wth d T (u, v) d T (x, y)+k. Part (1) of Observaton 2 ndcates a way to force an edge xy to be a tree edge, or to force a path of the tree to belong to certan part of the graph: Choosng

6 168 A.Brandstädt et al. H x e y Fg. 3. The graph obtaned from H and S k [x, y] by dentfyng the edge e = xy k = t 1 shows that xy must be an edge of T, or else the xy-path n T must belong to the part S t 1 [x, y]. We now descrbe the reducton. Let F be a 3SAT formula wth m clauses C j =(c j1,c j2,c j3 ) over n varables v. Set l := t 2 2 and λ := l 2. Snce t 4, l 0 and λ 0. For each varable v create the graph G(v ) as follows. Set q 0 := v, q l+1 := v.we wll use u {q 0,q l+1 } as a vertex n our graph as well as a lteral n the gven 3SAT formula F. Take a clque Q on l + 2 vertces q 0,...,q l,q l+1. For each edge xy {q k q k+1 : 0 k l} create an S t 1[x, y].no two of the S t 1 have a vertex n common unless those n {x, y}. Take a chordless path on vertces s 0,s 1,...,s λ and edges s k s k+1,0 k<λ. Connect each s k,0 k λ, to exactly q 0 and q l+1. For each edges xy {s k s k+1 : 0 k<λ} create an S t 2[x, y]. For each edges xy {s 0 q 0,s 0 q l+1,s λ q 0,s λ q l+1 } create an S t (l+2) [x, y]. Note that the clque Q s a cutset of G(v ) and the components of G(v ) Q are chordal. Thus, G(v ) s a chordal graph. See also Fgure 4. For each clause C j create the graph G(C j ) as follows. If t s even, G(C j )s smply a sngle vertex a j.ift s odd, G(C j ) s the graph S t 1 [a 1 j,a2 j ]. In any case, G(C j ) s a chordal graph. Fnally, the graph G = G(F ) s obtaned from all G(v ) and G(C j ) by dentfyng all vertces s 0 to a sngle vertex s, and addng the followng addtonal edges: connect every vertex n Q wth every vertex n Q,. Thus, the clques Q,1 n, form together a clque Q n G, for each lteral u {q 0,ql+1 }, fu C j then connect u wth a j, respectvely, wth a 1 j and a2 j, accordng to the party of t. The descrpton of the graph G = G(F ) s complete. Clearly, G can be constructed n polynomal tme. { t +1f t s even, Lemma 1 G s chordal, and dam(g) t +2f t s odd.

7 Tree Spanners on Chordal Graphs 169 Fg. 4. The graph G(v ) Lemma 2 Suppose G admts a tree t-spanner. Then F s satsfable. Lemma 3 Suppose F s satsfable. Then G admts a tree t-spanner. Theorem 1 follows from Lemmas 1-3. We remark that the chordal graph G constructed above always admts a tree (t + 1)-spanner. 3 Tree Spanners n Chordal Graphs of Smaller Dameter It s known [8,9] that for a chordal graph G, dam(g) 2r(G) 2 holds. Ths already yelds the followng. Theorem 5 Let t 2 be an even nteger. Every chordal graph of dameter at most t 1 admts a tree t-spanner, and such a tree spanner can be constructed n lnear tme. We remark that there are chordal graphs of dameter t wthout tree t-spanner. Thus, Theorem 5 s best possble under dameter constrants. Corollary1 Every chordal graph of dameter at most 3 has a tree 4-spanner, and such a tree spanner can be constructed n lnear tme. It remans an nterestng open queston whether exstence of a tree 3-spanner n a gven chordal graph of dameter at most 3 can be tested n polynomal tme. Lemma 4 Every chordal graph G admts a tree (2r(G))-spanner, and such a tree spanner can be constructed n lnear tme.

8 170 A.Brandstädt et al. Let now t be an odd nteger (t 3). From Lemma 4 and the fact that 2r(G) dam(g) 2r(G) 2 holds for any chordal graph G, we mmedately deduce. Theorem 6 Every chordal graph of dameter at most t 2 admts a tree t- spanner, and such a tree spanner can be constructed n lnear tme. It would be nce to show also that, f t 3 s an odd nteger, then every chordal graph of dameter at most t 1 admts a tree t-spanner. But, although for chordal graphs wth dam(g) 2r(G) 1 ths s true, t fals to hold for chordal graphs of dameter 2r(G) 2. There are even chordal graphs of dameter 2 wthout tree 3-spanners. In what follows we wll show that the exstence of a tree (2r(G) 1)-spanner n a chordal graph of dameter 2r(G) 2 depends on the exstence of a tree 3-spanner n a chordal graph of dameter 2. Frst we present some auxlary results. A subset S V s m convex f S contans every vertex on every chordless path between vertces of S. For a subset S V and a vertex v V, let proj(v, S) ={u S : d G (v, u) =d G (v, S)} be the metrc projecton of v to S. For a subset X V, let proj(x, S) = v X proj(v, S). A subset A V s a two-set n G f d G(v, u) 2 holds for every v, u A. Lemma 5 Let G be a (not necessarly chordal) graph. The metrc projecton proj(a, S) of any two-set A to an m convex set S of G s a two-set. Lemma 6 In every chordal graph G =(V,E) of dameter 2r(G) 2 there exsts a two-set S such that d G (v, S) r(g) 2 for every v V. Moreover such a two-set can be determned wthn tme O( V 3 ). Lemma 7 Every maxmal by ncluson two-set of a chordal graph s m-convex. Theorem 7 Chordal graphs of dameter 2r(G) 2 admt tree (2r(G) 1)- spanners f and only f chordal graphs of dameter 2 admt tree 3-spanners. Moreover, f a tree 3-spanner of any chordal graph of dameter 2 can be found n polynomal tme, then a tree (2r(G) 1)-spanner of a chordal graph of dameter 2r(G) 2 can be found n polynomal tme, too. We do not know how to use ths theorem for general chordal graphs (snce not all chordal graphs of dameter 2 have tree 3-spanners), but ths theorem could be very useful for those heredtary subclasses of chordal graphs where each graph of dameter 2 s tree 3-spanner admssble. Then, for every graph of dameter at most t 1 from those classes, a tree t-spanner wll exst and t could be found n polynomal tme f correspondng tree 3-spanner s constructable n polynomal tme. For an arbtrary chordal graph G wth dam(g) =2r(G) 2, t can happen that a chordal graph of dameter at most 2, generated by a two-set of G (found as descrbed n Lemma 6 and Theorem 7), does not have a tree 3-spanner, but yet G tself admts a (2r(G) 1)-spanner. We are stll workng on Tree (2r(G) 1)- Spanner problem n chordal graphs of dameter 2r(G) 2. It s natural to ask whether a combnaton of Theorem 7 and Theorem 9 works.

9 Tree Spanners on Chordal Graphs Tree 3-Spanners n Chordal Graphs of Dameter 2 In ths secton, we gve an applcaton of Theorem 7 as well as a crteron for the tree 3-spanner admssblty of chordal graphs of dameter at most 2. A graph G s non-trval f t has at least one edge. Lemma 8 Let G be a non-trval chordal graph of dameter at most 2. If G does not contan a clque K 5 on fve vertces, then G has a domnatng edge,.e., an edge e E such that d G (v, e) 1 for any v V. Snce nether planar graphs nor 3-trees have clques on 5 vertces and any graph wth a domnatng edge s trvally tree 3-spanner admssble, we conclude. Corollary2 Let G be a non-trval graph of dameter at most 2. IfG s a planar chordal graph or a k-tree for k 3, then G has a domnatng edge and hence a tree 3-spanner. As we mentoned n ntroducton, there s no constant t such that planar chordal graphs or k-trees (k 2) are tree t-spanner admssble. So, t s nterestng to menton the followng result. Theorem 8 Every chordal graph of dameter at most t 1, f t s planar or a k-tree (k 3), has a tree t-spanner and such a tree spanner can be constructed n polynomal tme. In what follows we wll assume that G s an arbtrary chordal graph whch admts a tree 3-spanner T. Note that any tree of dameter at most 2 s a star and any tree of dameter 3 has a domnatng edge (n ths case T s called a bstar). Lemma 9 For any maxmal (by ncluson) clque C of G one of the followng condtons holds. a) C nduces a star n T, b) ether C nduces a bstar n T or there s a vertex v/ C such that C {v} nduces a bstar n T. Clearly, T s a star only f G has an unversal vertex, and the dameter of T s 3 only f G has a domnatng edge. The followng theorem handles the case of all chordal graphs of dameter at most 2. Unfortunately, not every such graph has a domnatng edge. There are chordal graphs of dameter 2 whch do not have any tree 3-spanners, and there are chordal graphs of dameter 2 that have a tree 3-spanner but all those spanners are of dameter 4. Theorem 4 wll follow from ths theorem. Theorem 9 A chordal graph G of dameter at most 2 admts a tree 3-spanner f and only f there s a vertex v n G such that any connected component of the second neghborhood of v has a domnatng vertex n N(v).

10 172 A.Brandstädt et al. 5 Tree Spanners n Strongly Chordal Graphs and k-splt Graphs For an nteger k 3, a k-sun conssts of a k-clque {v 1,...,v k } and a k-vertex stable set {u 1,...,u k }, and edges u v,u v +1,1 <k, and u k v k,u k v 1.A chordal graph s strongly chordal [11] f t does not contan a k-sun as an nduced subgraph. In [3], t s proved that every strongly chordal graph admts a tree 4- spanner and such a tree spanner can be constructed n lnear tme. Not every strongly chordal graph has a tree 3-spanner. Actually, Tree 3-Spanner remans open on strongly chordal graphs. A k-planet s obtaned from a k-path v 1 v 2 v 3 v k and a trangle abc by addng edges av,1 k 1and bv,2 k;see Fgure k 1 k Fg. 5. A k-planet Defnton 1 A chordal graph s called very strongly chordal f t does not contan a k-planet as an nduced subgraph. As a 3-sun s a 3-planet and every k-sun (k 4) contans an nduced 4-planet, the class of very strongly chordal graphs s properly contaned n the class of strongly chordal graphs. Moreover, the class of very strongly chordal graphs contans all nterval graphs and all dstance heredtary chordal graphs, called ptolemac graphs [15]. The nce feature of ths subclass of strongly chordal graphs s Theorem 10 Every very strongly chordal graph admts a tree 3-spanner and such a tree spanner can be constructed n lnear tme. Another well-known subclass of strongly chordal graphs conssts of the ntersecton graphs of drected paths n a rooted drected tree, called drected path graphs. The class of drected path graphs generalzes nterval graphs naturally, and contans all ptolemac graphs, and s tree 3-spanner admssble [17]. The ntersecton graphs of paths n a tree are called (undrected) path graph. We call shortly a graph strongly path graph f t s strongly chordal as well as a path graph. Clearly, every drected path graph s a strongly path graph, but not vce versa. Indeed, there are many strongly path graphs havng no tree 3-spanner (whle every drected path graph does [17]). Moreover, n contrast to strongly chordal graphs, for every t, there s a path graph havng no tree t-spanner [16]. A splt graph s one whose vertex set can be parttoned nto a clque and a stable set. Splt graphs are exactly those chordal graphs whose complements are chordal as well. It s known (and easy to see;cf. [5,16,25]) that every splt graph admts a tree 3-spanner. We are gong to descrbe a new subclass of chordal graphs contanng all splt graphs and stll are tree 3-spanner admssble.

11 Tree Spanners on Chordal Graphs 173 Frst, for an arbtrary graph G let S(G) be the set of all smplcal vertces of G. We also use S(G) for the subgraph of G nduced by S(G). Lemma 10 If G \ S(G) hasatree(t 1)-spanner then G hasatreet-spanner. Defnton 2 For an arbtrary graph G and an nteger k 0 let G k := G k 1 \ S(G k 1 ); G 0 := G. AgraphG s called k-splt f G k s a clque. Clearly, 0-splt graphs are exactly the clques, and all splt graphs are 1-splt but not vce versa. The followng fact s probably known. Proposton 2 A graph G s chordal f and only f G s k-splt for some k. Theorem 11 Every k-splt graph admts a tree (k +2)-spanner. Corollary3 All 1-splt graphs, hence all splt graphs, admt a tree 3-spanner, and a such a tree 3-spanner can be constructed n lnear tme, gven the set of all smplcal vertces. Note that the exstence of a tree (k+2)-spanner n k-splt graphs s best possble: there are many k-splt graphs wthout tree (k+1)-spanner;for example, the 3-sun s 1-splt (even splt) and has no tree 2-spanner. 6 Concluson In ths paper we have proved that, for any t 4, Tree t-spanner s NPcomplete on chordal graphs of dameter at most t + 1 (f t s even), respectvely, at most t+2 (f t s odd), mprovng the hardness result n [6] on a restrcted wellunderstood graph class. We have shown that every chordal graph G of dameter at most t 1 s tree t-spanner admssble f dam(g) 2r(G) 2. The complexty of Tree t-spanner remans unresolved on chordal graphs of dameter t (f t s even) and of dameter t or t + 1 (f t s odd). Tree t- Spanner remans also open on path graphs and the case t = 3 remans even open on path graphs that are strongly chordal graphs as well. However, we have shown that all very strongly chordal graphs, a subclass of strongly chordal graphs that contans all nterval graphs and all ptolemac graphs, are tree 3-spanner admssble, and a tree 3-spanner for a gven very strongly chordal graph can be constructed n lnear tme. Ths mproves known results on tree 3-spanners n nterval graphs [16,18,23]. We have also mproved known results on tree 3- spanners n splt graphs [5,16,25] by showng that all 1-splt graphs, a subclass of chordal graphs contanng all splt graphs, are tree 3-spanner admssble, and a tree 3-spanner for a 1-splt graph can be constructed n lnear tme, gven the set of ts smplcal vertces. We presented a polynomal tme algorthm for the Tree 3-Spanner problem on chordal graphs of dameter at most 2. Many questons reman stll open. Among them: (1) Can Tree 3-Spanner be decded effcently on (strongly) chordal graphs? (2) Can Tree (2r(G) 1)-Spanner be decded effcently on chordal graphs of dameter 2r(G) 2? (3) What s the complexty of Tree t-spanner for chordal graphs of dameter at most t?.

12 174 A.Brandstädt et al. References 1. B.Awerbuch, A.Baratz, D.Peleg, Effcent broadcast and lght-weghted spanners, manuscrpt, H.-J.Bandelt, A.Dress, Reconstructng the shape of a tree from observed dssmlarty data, Adv. Appl. Math. 7 (1986) A.Brandstädt, V.Chepo, F.Dragan, Dstance approxmatng trees for chordal and dually chordal graphs, J. Algorthms 30 (1999) A.Brandstädt, V.B.Le, J.Spnrad, Graph Classes: A Survey, SIAM Monographs on Dscrete Math. Appl., (SIAM, Phladelpha, 1999) 5.L.Ca, Tree spanners: Spannng trees that approxmate the dstances, Ph.D. thess, Unversty of Toronto, L.Ca, D.G.Cornel, Tree spanners, SIAM J. Dscrete. Math. 8 (1995) L.Ca, D.G.Cornel, Tree spanners: An overvew, Congressus Numer. 88 (1992) G.J. Chang, G.L. Nemhauser, The k-domnaton and k-stablty problems on sunfree chordal graphs, SIAM. J. Alg. Dsc. Meth. 5 (1984) V.D.Chepo, Centers of trangulated graphs, Math. Notes 43 (1988) V.D. Chepo, F.F. Dragan, Lnear-tme algorthm for fndng a center vertex of a chordal graph, Lecture Notes n Computer Scence 855 (1994) M.Farber, Characterzatons of strongly chordal graphs, Dscrete Math. 43 (1983) S.P.Fekete, J.Kremer, Tree spanners n planar graphs, Dscrete Appl. Math. 108 (2001) F.Gavrl, The ntersecton graphs of subtrees n trees are exatly the chordal graphs, J. Combn. Theory (B) 16 (1974) M.C.Golumbc, Algorthmc Graph Theory and Perfect Graphs (Academc Press, New York, 1980) 15.E.Howorka, A characterzaton of ptolemac graphs, J. Graph Theory 5 (1981) Hoàng-Oanh Le, Effzente Algorthmen für Baumspanner n chordalen Graphen, Dploma thess, Dept.of mathematcs, techncal unversty of Berln, H.-O Le, V.B. Le, Optmal tree 3-spanners n drected path graphs, Networks 34 (1999) M.S.Madanlal, G.Venkatesan, C.Pandu Rangan, Tree 3-spanners on nterval, permutaton and regular bpartte graphs, Inform. Process. Lett. 59 (1996) I.E.Papoutsaks, Two structure theorems on tree spanners, M.Sc. thess, Dept.of Computer Scence, Unversty of Toronto, I.E.Papoutsaks, On the unon of two tree spanners of a graph, Preprnt, D.Peleg, Dstrbuted Computng: A Localty-Senstve Approach, SIAM Monographs on Dscrete Math. Appl., (SIAM, Phladelpha, 2000) 22. D.Peleg, A.Schaeffer, Graph spanners, J.Graph Theory 13 (1989) E.Prsner, Dstance approxmatng spannng trees, n: Proc.STACS 97, Lecture Notes n Computer Scence, Vol (Sprnger, Berln, 1997) J.Soares, Graph spanners: A survey, Congressus Numer. 89 (1992) G.Venkatesan, U.Rotcs, M.S.Madanlal, J.A.Makowsky, C.Pandu Ragan, Restrctons of mnmum spanner problems, Informaton and Computaton 136 (1997)