On collinear sets in straight line drawings

ON COLLINEAR SETS I N STRAIGHT LINE DRA WINGS ALEXANDER RA VSKY ∗ AND OLEG VERBITSKY ∗ † Abstract. W e consider straight line drawings of a pla nar gra ph G with pos sible edge crossing s . The untangling pr oblem is to eliminate all edge cross ing s by moving as few vertices as po ssible to new po sitions. Let fix ( G ) deno te the max im um nu m be r of vertices that can b e left fixe d in the w o rst case. In the al lo c ation pr oblem , w e are given a planar graph G on n v ertices together with an n -p oint set X in the plane and ha ve to dra w G without edge crossing s so that as many vertices as po s sible a r e lo cated in X . Let fit ( G ) denote the maximum num b er of po int s fitting this pur p o s e in the w or st case. As fi x ( G ) ≤ fit ( G ), we are interested in uppe r b ounds f o r the latter and low er b ounds for the for mer parameter. F or each ǫ > 0, we construct an infinite sequence of graphs with fi t ( G ) = O ( n σ + ǫ ), where σ < 0 . 9 9 is a known graph-theor etic constant, namely the short- ness exp onent for the clas s o f cubic polyhedr al graphs. T o the b est of our knowl- edge, this is the first exa mple of gr aphs w ith fit ( G ) = o ( n ). On the other ha nd, we prove that fix ( G ) ≥ p n/ 30 for all G with tree-width a t most 2. This extends the lower b ound obtained b y Goao c e t al. [ Discr et e and Computational Ge ometry 42:542 –569 (200 9)] fo r outerplanar graphs. Our upp er b o und for fit ( G ) is ba sed on the fact that the constr ucted g raphs can hav e only few collinear vertices in any cr o ssing-free dr awing. T o prov e the low er bound for fix ( G ), w e show that graphs of tree-width 2 admit drawings tha t hav e large sets of collinear vertices with s ome additional sp ecial prop erties. 1. Intr oduction 1.1. Basic definitions. Let G be a planar graph. The ve rtex set o f G will be denoted b y V G . The letter n will b e reserv ed to alw a ys denote the n umber of v ertices in V G . By a dr awing of G w e mean an arbitrar y injectiv e ma p π : V G → R 2 . The p oints in π ( V G ) will b e referred to as vertic es of the dra wing. F or an edge uv of G , t he segmen t with endp oints π ( u ) and π ( v ) will b e referred to as an e dge of the drawing. Th us, w e a lw ay s consider str ai g ht-line dra wings. It is quite p ossible that in π w e encoun ter edge crossings and ev en o v erlaps. A dra wing is pla ne (o r cr ossing-fr e e ) if this do es not happ en. Giv en a drawin g π of G , define fix ( G, π ) = m ax π ′ plane | { v ∈ V G : π ′ ( v ) = π ( v ) } | . ∗ Institute fo r Applied Pro blems of Mechanics and Mathematics, Nauko v a St. 3- B , Lviv 79 0 60, Ukraine. † Current addr e s s: Hum b oldt-Universit¨ at zu Ber lin, Institut f ¨ ur Informatik, Un ter den Linden 6, D-10099 Berlin. Supp orted by the Alexander von Humboldt F oundatio n. 1 2 ALEXANDER RA VSKY and OLEG VERBITSKY Giv en a n n -p oint set X in the plane, let fix X ( G ) = min π : π ( V G )= X fix ( G, π ) . F urthermore, w e define fix ( G ) = min X fix X ( G ) = min π fix ( G, π ) . (1) In other w ords, fix ( G ) is the maximum n umber of v ertices which can b e fixed in any dra wing of G while untangli n g it. Giv en an n -p o in t set X , consider no w a related parameter fit X ( G ) = max π plane | π ( V G ) ∩ X | . In words, if w e wan t to draw G allo cating its ve r tices at p oin ts of X , then fit X ( G ) tells us ho w many p o in ts of X can fit for this purp ose. T o analyze the al lo c ation problem in the w orst case, w e define fit ( G ) = min X fit X ( G ) . Note that fit X ( G ) = max π : π ( V G )= X fix ( G, π ). It follow s that fix X ( G ) ≤ fit X ( G ) and, therefore, fix ( G ) ≤ fit ( G ). 1.2. Kno wn r esults on the un tangling problem. No efficien t w a y fo r ev aluating the parameter fix ( G ) is know n. Note that computing fix ( G, π ) is NP-hard [10, 16]. Essen tial efforts are needed to estimate fix ( G ) ev en for cycles, for whic h w e know b ounds 2 − 5 / 3 n 2 / 3 − O ( n 1 / 3 ) ≤ fix ( C n ) ≤ O (( n log n ) 2 / 3 ) due to, resp ectiv ely , Cibulk a [5] a nd P a ch a nd T ardos [1 5]. In the general case Bose et al. [3] establish a low er b o und fix ( G ) ≥ ( n/ 3) 1 / 4 . (2) A b etter b ound fix ( G ) ≥ p n/ 2 (3) is pro v ed for a ll trees ( Bo se et a l. [3 ], Goao c et al. [10]) a nd, more generally , outer- planar graphs (G o ao c et al. [1 0 ], cf. Corollary 4 .4 b elo w). On the other hand, [3, 10, 14] pro vide examples of planar graphs (ev en acyclic ones) with fix ( G ) = O ( √ n ) . (4) In particular, for the fan gra phs F n w e ha ve fix X ( F n ) ≤ (2 √ 2 + o (1)) √ n for ev ery X , (5) see [14]. Cibulk a [5] establishes s ome general upp er b ounds, namely fix ( G ) = O ( √ n (log n ) 3 / 2 ) for graphs whose maxim um degree and dia meter are b ounded b y a logarithmic function and fix ( G ) = O (( n log n ) 2 / 3 ) for 3-connected graphs. ON COLLIN EAR SETS I N STRA IGHT LINE DR A WI NGS 3 1.3. Kno wn r esults on t he allo cation problem. The question whether or not fit X ( G ) = n has b een studied in the lit era t ur e, esp ecially for X in general p o sition. If X is in con vex p osition, an y t r iangulation on X is outerplanar. By this r eason, fit X ( G ) < n for all non-o uterplana r graphs G and all sets X in conv ex p osition. On the o ther hand, in [11] it is pro v ed that fit X ( G ) = n for a ll outerplanar G and all X in general p osition. Other results and references on this sub ject can b e found, e.g., in [8]. It is kno wn that there are Ω(27 . 2 2 n ) unlab eled pla na r graphs with n v ertices [9], while a set of n p oin ts in con v ex p osition admits no more t ha n O (11 . 66 n ) plain dra wings [7]. Comb ining the tw o r esults, w e see that, if X is in conv ex p osition, then fit X ( G ) < n for almost all planar G . 1.4. Our present con tr ibution. W e aim at prov ing upp er b ounds for fit ( G ) and lo we r b ounds for fix ( G ). O ur approac h to b o th problems is based on analysis of collinear sets of ve rtices in straight line graph dra wings. W e show the relev ance o f the follow ing questions. Ho w man y collinear v ertices can o ccur in a plane dra wing of a graph G ? If there is a larg e collinear set, which useful features can it hav e? Supp ose that π is a crossing-free dra wing of a graph G . A set o f v ertices S ⊆ π ( V G ) in π is c ol line ar if all of them lie on a line ℓ . By a c onformal displac ement of S we mean a r elo cation δ : S → ℓ preserving the relativ e order in whic h the ve r t ices in S lie in ℓ . W e call S fr e e if eve r y conformal displacemen t δ : S → ℓ is extendable to a mapping δ : π ( V G ) → R 2 so that δ ◦ π is a crossing-free draw ing of G (i.e., whenev er w e shift vertices in S alo ng ℓ without breaking their relativ e order, then all edge crossings that may arise can b e eliminated b y subsequen tly mo ving the ve r tices in π ( V G ) \ S ). Let ¯ v ( G, π ) denote the maxim um size o f a collinear set in π a nd ˜ v ( G, π ) the maxim um size o f a free collinear set in π . Define ¯ v ( G ) = max π plane ¯ v ( G, π ) and ˜ v ( G ) = max π plane ˜ v ( G, π ) . Ob viously , ˜ v ( G ) ≤ ¯ v ( G ). These parameters hav e a direct relation to fix ( G ) and fit ( G ), namely p ˜ v ( G ) ≤ fix ( G ) ≤ fit ( G ) ≤ ¯ v ( G ) . (6) The latter inequalit y follow s immediately from the definitions. The first inequalit y is pro v ed as Theorem 4.1 b elow . In Section 3 w e construct, for eac h ǫ > 0, an infinite sequence o f graphs with ¯ v ( G ) = O ( n σ + ǫ ) where σ ≤ log 22 log 23 is a kno wn graph-theoretic constan t , namely the shortness exp onen t for the class of cubic p olyhedral graphs (see Section 2 for the definition). T o the b est of our kno wledge, this giv es us the first example of graphs with fit ( G ) = o ( n ). While the know n upp er b ounds (4) for fix ( G ) are still b etter, note that the problems o f b ounding fix ( G ) and fit ( G ) from ab ov e are inequiv a lent. The t w o parameters can b e f a r a w a y from one another: for example, in contrast with (5) w e hav e fit X ( F n ) ≥ n − 1 for any X . By the lo w er bound in (6), w e ha v e fix ( G ) = Ω( √ n ) whenev er ˜ v ( G ) = Ω( n ). Therefore, iden tification of classes of planar graphs with linear ˜ v ( G ) is of big interes t. In Section 4 w e sho w that ˜ v ( G ) ≥ n/ 2 fo r ev ery outerplanar graph G . This giv es us 4 ALEXANDER RA VSKY and OLEG VERBITSKY another pro of of the b ound fix ( G ) ≥ p n/ 2 prov ed f o r outerplanar graphs b y Goao c et al. [10] 1 . F urthermore, w e consider t he broader class of graphs with tree-width at mo st 2. It coincides with the class of partial 2-trees and contains also all series-parallel graphs (see, e.g. [2, Sect. 8.3 ]). F or an y graph G in this class, w e pro v e that ˜ v ( G ) ≥ n/ 30 and, therefore, fix ( G ) ≥ p n/ 30. The pro of of this result t ak es Section 5. Note that the sublinear upp er b ound for ¯ v ( G ) is established in Section 3 for a sequence of graphs whose tree-width is b ounded b y a constan t. W e conclude with the discussion of op en pro blems in Section 6. 2. Preliminaries Giv en a planar graph G , w e denote the num b er of vertices , edges, and faces in it, resp ectiv ely , b y v ( G ), e ( G ), and f ( G ). The latt er n umber do es not dep end on a particular plane embedding of G and hence is w ell defined. Moreo v er, fo r connected G w e ha ve v ( G ) − e ( G ) + f ( G ) = 2 (7) b y Euler’s formu la. A graph is k -c onne cte d if it has more than k v ertices and stays connected af- ter remo v al of any k vertice s. 3- connected planar graphs are called p olyhe dr al as, according t o Steinitz’s theorem, these gra phs are exactly the 1-sk eletons o f conv ex p olyhedra. By Whitney’s theorem, all plane em b eddings of a p o lyhedral gr a ph G are equiv alen t, t ha t is, obta inable fr om o ne ano t her by a plane homeomorphism up to the c hoice of out er face. In particular, the set of facial cycles (i.e., b oundaries of faces) of G do es not dep end on a par t icular pla ne em b edding. A planar g r a ph G is maximal if adding an edge b et w een an y tw o non-a djacen t v ertices of G violates plana r ity . Maximal planar g raphs on more than 3 v ertices are 3- connected. Clearly , all facial cycles in suc h gra phs ha ve length 3. By this reason ma ximal planar g raphs a re a lso called triang ulation s . Note that for ev ery triangulation G w e hav e 3 f ( G ) = 2 e ( G ). Com bined with (7), this giv es us f ( G ) = 2 v ( G ) − 4 . (8) The dual of a p olyhedral graph G is a gr aph G ∗ whose v ertices are the faces of G (r epresen ted b y their facial cycles). Tw o faces are adjacen t in G ∗ iff they share a common edge. G ∗ is also a p olyhedral gra ph. If w e consider ( G ∗ ) ∗ , we obta in a graph isomorphic to G . In a c ubic graph ev ery v ertex is inciden t to exactly 3 edges. As easily seen, the dual of a triangulation is a cubic graph. Con v ersely , the dua l of an y cubic p olyhedral graph is a triangulation. The cir cumfer enc e of a gra ph G , denoted b y c ( G ), is the length of a longest cycle in G . The shortness exp onen t of a class of graphs G is t he limit inferior of quotients log c ( G ) / log v ( G ) o ver all G ∈ G . Let σ denote the shortness exp onen t for the class 1 A prelimina ry version of [1 0] gave a so mew ha t worse b ound of fix ( G ) ≥ p n/ 3. An improvemen t to p n/ 2 was made in the early version of the pr esent pap er indep endently of [10]. ON COLLIN EAR SETS I N STRA IGHT LINE DR A WI NGS 5 Figure 1. An example of the construction: G 1 = K 4 , G 2 , G 3 . of cubic p olyhedral graphs. It is know n that 0 . 753 < σ ≤ log 22 log 23 = 0 . 985 . . . (see [1] for the lo wer b ound and [1 3, Theorem 7(iv)] for the upp er b ound). 3. Graphs with s mall collinear sets W e here construct a sequence o f triangulations G with ¯ v ( G ) = o ( v ( G )). F or our analysis we will need another parameter of a straigh t line dra wing. Giv en a crossing- free drawing π of a graph G , let ¯ f ( G, π ) denote the maxim um n umber of collinear p oin ts in the plane suc h that eac h of them is an inner p oin t of some face of π and no t wo of them are in the same face. Let ¯ f ( G ) = max π ¯ f ( G, π ). In other words, ¯ f ( G ) is equal to the maximum num b er of faces in some stra igh t line drawing of G whose interiors can b e cut by a line. F urther on, sa ying that a line cuts a fa ce, w e mean that the line inte rsects the in terior of this fa ce. F or the triangulat io ns constructed b elo w, we will show that ¯ v ( G ) is small with resp ect to v ( G ) b ecause ¯ f ( G ) is small with resp ect to f ( G ) (though w e do not know an y r elat io n b et w een ¯ v ( G ) and ¯ f ( G ) in general). Our construction can b e thought of as a recursiv e pro cedure for essen tially decreasing the ratio ¯ f ( G ) /f ( G ) at eac h recursion step provide d tha t w e initially ha ve ¯ f ( G ) < f ( G ). Starting from an arbitrary triangulation G 1 with at least 4 v ertices, w e recursiv ely define a sequence of tria ng ulations G 1 , G 2 , . . . . T o defi ne G k , w e will desc rib e a spherical dra wing δ k of this graph. Let δ 1 b e an arbitr ary dra wing of G 1 on a sphere. F urthermore, δ i +1 is obtained from δ i b y triangula t ing each f a ce of δ i so that this triangulation is isomorphic to G 1 . An example is sho wn on Fig. 1. In general, upgrading δ i to δ i +1 can b e done in differen t w ay s, that may lead to non-isomorphic v ersions o f G i +1 . W e mak e an ar bitr ary c hoice and fix the result. Lemma 3.1. De n ote f = f ( G 1 ) , ¯ f = ¯ f ( G 1 ) , and α = log( ¯ f − 1) log( f − 1) . 1. f ( G k ) = f ( f − 1 ) k − 1 . 2. ¯ f ( G k ) ≤ ¯ f ( ¯ f − 1) k − 1 . 6 ALEXANDER RA VSKY and OLEG VERBITSKY 3. ¯ v ( G k ) < c v ( G k ) α , wher e c is a c onstant dep ending only on G 1 . Pr o of. The first part follows fro m the obvious recurrence f ( G i +1 ) = f ( G i )( f − 1) . W e ha ve to prov e the o t her tw o pa rts. Consider an arbit r a ry crossing-free straigh t line drawing π k of G k . Recall that, b y construction, G 1 , . . . , G k − 1 is a chain of subgra phs of G k with V G 1 ⊂ V G 2 ⊂ . . . ⊂ V G k − 1 ⊂ V G k . Let π i b e the par t of π k whic h is a drawing of the subgra ph G i . By the Whitney theorem, π k can b e obtained fr o m δ k (the spherical draw ing defining G k ) b y a n appropriate stereographic pro jection of the sphere to the plane com bined with a homeomorphism of the plane onto itself. It fo llo ws that, lik e δ i +1 and δ i , drawings π i +1 and π i ha ve the f o llo wing prop erty : the restriction of π i +1 to an y face of π i is a dra wing of G 1 . Given a fa ce F of π i , the restriction of π i +1 to F (i.e., a plane graph isomorphic to G 1 ) will b e denoted b y π i +1 [ F ]. Consider now an arbitrary line ℓ . Let ¯ f i denote the num b er of faces in π i cut by ℓ . By definition, we hav e ¯ f 1 ≤ ¯ f . (9) F or each 1 ≤ i < k , we ha v e ¯ f i +1 ≤  ¯ f if ¯ f i = 1 , ¯ f i ( ¯ f − 1) if ¯ f i > 1 . (10) Indeed, let K denote the outer f a ce of π i . Equalit y ¯ f i = 1 means that, of all faces of π i , ℓ cuts o nly K . Within K , ℓ can cut only faces of π i +1 [ K ] and, therefore, ¯ f i +1 ≤ ¯ f . Assume that ¯ f i > 1. Within K , ℓ can no w cut at most ¯ f − 1 faces of π i +1 (b ecause ℓ cuts R 2 \ K , a face of π i +1 [ K ] outside K ). Within any inner face F of π i , ℓ can cut at most ¯ f − 1 faces of π i +1 (the subtrahend 1 corresp onds to the outer face of π i +1 [ F ], whic h surely con tributes to ¯ f but is outside F ). The n um b er of inner faces F cut b y ℓ is equal to ¯ f i − 1 (again, the subtrahend 1 corresp onds to the o uter f ace of π i ). W e therefore hav e ¯ f i +1 ≤ ( ¯ f − 1) + ( ¯ f i − 1)( ¯ f − 1) = ¯ f i ( ¯ f − 1), completing the pro of of (10). Using (9) a nd ( 1 0), a simple inductiv e argument gives us ¯ f i ≤ ¯ f ( ¯ f − 1) i − 1 (11) for each i ≤ k . As π k and ℓ a re ar bitrary , part 2 of t he lemma is prov ed by setting i = k in (11). T o prov e part 3, we ha v e to estimate from ab ov e ¯ v = | ℓ ∩ V ( π k ) | , the n um b er of v ertices of π k on the line ℓ . Put ¯ v 1 = | ℓ ∩ V ( π 1 ) | and ¯ v i = | ℓ ∩ ( V ( π i ) \ V ( π i − 1 )) | fo r 1 < i ≤ k . Clearly , ¯ v = P k i =1 ¯ v i . Abbreviate v = v ( G 1 ). It is easy t o see that ¯ v 1 ≤ v − 2 and, for all 1 < i ≤ k , ¯ v i ≤ ¯ f i − 1 ( v − 3) . ON COLLIN EAR SETS I N STRA IGHT LINE DR A WI NGS 7 It f ollo ws that ¯ v ≤ ( v − 2 ) + ( v − 3) k − 1 X i =1 ¯ f i ≤ ( v − 3) ¯ f ¯ f − 2 ( ¯ f − 1) k − 1 , (12) where w e use ( 1 1) for the latter estimate. It remains to express the obtained b ound in terms o f v ( G k ). By (8) and by part 1 of the lemma, w e hav e ( f − 1) k − 1 < 2 v ( G k ) /f and, therefore, ( ¯ f − 1) k − 1 = ( f − 1) α ( k − 1) < (2 /f ) α v ( G k ) α . Plugging this in to (12), we arriv e at the desired b ound f or ¯ v and hence for ¯ v ( G k ).  W e no w need an initial triangulation G 1 with ¯ f ( G 1 ) < f ( G 1 ). The follo wing lemma sho ws a direction where one can seek for suc h tr ia ngulations. Lemma 3.2. F o r ev e ry triangulation G with mor e than 3 vertic es, w e have ¯ f ( G ) ≤ c ( G ∗ ) . Pr o of. Given a crossing-free dra wing π of G and a line ℓ , w e ha v e to sho w that ℓ crosses no mo r e tha n c ( G ∗ ) faces o f π . Shift ℓ a little bit to a new p osition ℓ ′ so that ℓ ′ do es no t g o through any ve r t ex of π and still cuts all the fa ces that a re cut b y ℓ . Th us, ℓ ′ crosses b o undaries of fa ces o nly via inner p oints of edges. Eac h suc h crossing corresp onds to transition f r om o ne v ertex to another along an edge in t he dual gr a ph G ∗ . Not e that t his walk is b ot h started a nd finished at the outer fa ce of π . Since all faces are tria ng les, each of them is visited at mo st once. Therefore, ℓ ′ determines a cycle in G ∗ , whose length is at least the n umber of faces of π cut b y ℓ .  Lemma 3.2 suggests the follo wing c ho ice of G 1 : T ak e a cubic p olyhedral graph H approac hing the infim um of the set of quotients log ( c ( G ) − 1) / log( v ( G ) − 1) ov er all cubic p olyhedral graphs G a nd set G 1 = H ∗ . In particular, w e can appro a c h arbitrarily close to the shortness exp onent σ defined in Section 2. By Lemma 3.1.3, w e arr ive at the main result of this section. Theorem 3.3. L et σ denote the shortness exp onent of the class of c ubic p olyhe dr al gr aphs. Then for e ach α > σ ther e is a se quenc e of triangulations G with ¯ v ( G ) = O ( v ( G ) α ) . Corollary 3.4. F or infinitely many n ther e is a plana r gr aph G on n vertic es with fit ( G ) = O ( n 0 . 99 ) . Note that the graph G k constructed in the pr o of of Theorem 3.3 is obtained from G k − 1 in a n um b er o f steps by gluing with a cop y of G 1 at a clique of size 3. It follo ws that all G 1 , G 2 , . . . hav e equal t r ee-width. If w e tak e H to b e the Barnette- Bos´ ak-Lederb erg example of a non-Hamiltonia n cubic p olyhedral graph, then the construction star t s with G 1 = H ∗ of tree-width no more than 8. W e, therefore, obtain the following result. Corollary 3.5. F or some inte ger c onstant t ≤ 8 and r e al c onstant α ∈ (0 , 1) , ther e is a se quenc e of triangulations G of tr e e-width at most t such that fit ( G ) ≤ ¯ v ( G ) = O ( v ( G ) α ) . 8 ALEXANDER RA VSKY and OLEG VERBITSKY Remark 3.6. Theorem 3.3 can b e tra nslated to a result on con v ex p olyhedra. Give n a con v ex p o lyhedron π and a plane ℓ , let = v ( π , ℓ ) = V ( π ) ∩ ℓ where V ( π ) denotes the v ertex set of π . Giv en a p o lyhedral graph G , we define = v ( G ) = max π ,ℓ = v ( π , ℓ ) , where π ranges o v er con vex p o lyhedra with 1-sk eleton isomorphic to G . Using our construction of a sequenc e o f triangulatio ns G 1 , G 2 , G 3 , . . . , w e can prov e tha t = v ( G ) = O ( v ( G ) α ) for eac h α > σ and infinitely many p olyhedral graphs G . Gr ¨ un baum [12 ] inv estigated the minim um n umber of planes whic h are enough to cut all edges o f a con ve x p olyhedron π . Giv en a p o lyhedral graph G , define = e ( G ) = max π ,ℓ = e ( π , ℓ ) , where = e ( π , ℓ ) denotes the n um b er of edges that are cut by a plane ℓ in a con v ex p olyhedron π with 1- ske leton isomorphic to G . Using the relation = e ( G ) ≤ c ( G ∗ ), Gr ¨ un baum show ed (implicit in [12, pp. 893– 894]) tha t = e ( G ) = O ( v ( G ) β ) for eac h β > log 3 2 and infinitely man y G (where lo g 3 2 is the shortness exp onen t for the class of all p olyhedral graphs). 4. Graphs with large free collinear se ts Let π b e a crossing-free dra wing and ℓ b e a line. Recall that a set S ⊂ π ( V G ) ∩ ℓ is called fr e e if, whenev er w e displace the ve rtices in S along ℓ without violating their m utua l order, t hereby intro ducing edge crossings, w e are able to un ta ng le the mo dified dra wing by only mov ing the v ertices in π ( V G ) \ S . By ˜ v ( G ) w e denote the largest size of a free collinear set maximized ov er all dr awings of a graph G . Theorem 4.1. fix ( G ) ≥ p ˜ v ( G ) . Pr o of. Let fix − ( G ) b e defined similarly to (1) but with minimization ov er all collinear X (or ov er π suc h that π ( V G ) is collinear). Ob viously , fix ( G ) ≤ fix − ( G ). As prov ed in [14] (based on [3, Lemma 1]) , w e actually ha ve fix ( G ) = fix − ( G ) . (13) W e use this equality here. Supp ose tha t ( k − 1) 2 < ˜ v ( G ) ≤ k 2 . By (13), it suffices to show that a n y drawing π : V G → ℓ o f G on a line ℓ can b e made crossing-fr ee with kee ping k v ertices fixed. Let ρ b e a crossing-free dra wing of G suc h that, for some S ⊂ V G with | S | > ( k − 1) 2 , ρ ( S ) is a free collinear set on ℓ . By the Erd˝ os-Szek eres theorem, there exists a set F ⊂ S of k vertice s suc h that π ( F ) and ρ ( F ) lie on ℓ in the same order. By the definition of a fr ee set, there is a crossing-free drawing ρ ′ of G with ρ ′ ( F ) = π ( F ). Th us, w e can come fr o m π to ρ ′ with F staying fixed.  Theorem 4.1 sometimes give s a short w a y of prov ing b ounds of the kind fix ( G ) = Ω( √ n ). F o r example, for the wheel gra ph W n w e immediately obtain fix ( W n ) > √ n − 1 fro m an easy observ at ion that ˜ v ( W n ) = n − 2 (in fact, this rep eats the argumen t of Pac h and T ardos for cycles [15]). The classes of graphs with linear ON COLLIN EAR SETS I N STRA IGHT LINE DR A WI NGS 9 t 1 t 2 t 3 t 4 t 5 Figure 2. An outerplanar g raph and its tra c k dra wing. ˜ v ( G ) are therefore o f big in terest in the con text of disen tanglemen t of dra wings. One of suc h classes is addressed b elow. Giv en a dra wing π , we call it tr ack dr awing if there are parallel lines, called tr acks , suc h that ev ery v ertex of π lies on o ne of the lay ers and ev ery edge either lies on one of the lay ers o r connects endv ertices lying on tw o consecutiv e lay ers. W e call a graph tr ack dr awable if it has a crossing-free track draw ing. An obvious example of a trac k dra w able graph is a g r id graph P s × P s . It is also easy to see that an y tree is trac k draw a ble: t wo v ertices a re to b e aligned on the same lay er iff they are at the same distance from an arbit r a rily assigned ro ot. The latter example can b e considerably extended. Call a dra wing outerplanar if all the ve r t ices lie on the outer face. An outerplanar gr aph is a gr a ph admitting an outerplana r drawing (this definition do es not dep end on whether straight line o r curv ed dra wings are considered). The follo wing fact is illustrated b y Fig. 2 . Lemma 4.2 (F elsner, Liotta, and Wismath [6]) . Outerplanar gr aphs ar e tr ack dr aw- able. Lemma 4.3. F o r an y tr ack dr awab l e g r a ph G on n vertic es we hav e ˜ v ( G ) ≥ n/ 2 . Pr o of. Let π b e a trac k dra wing of G with trac ks t 1 , . . . , t s , lying in the plane in this order. It is practical to assume that t 1 , . . . , t s are parallel stra ig h t line segmen ts (rather than unbounded lines) con ta ining all the v ertices of π . Let ℓ b e a horizontal line. Consider tw o redraw ings of π . T o mak e a redra wing π ′ , w e put t 1 , t 3 , t 5 , . . . on ℓ one b y o ne. F or each ev en index 2 i , w e drop a p erp endicular p 2 i to ℓ b etw een t he segmen ts t 2 i − 1 and t 2 i +1 . W e then put each t 2 i on p 2 i so tha t t 2 i is in the upp er half-plane if i is o dd and in the lo wer half-plane if i is ev en. It is clear tha t suc h a relo cation can b e done so that π ′ is crossing-free (the whole pro cedure can b e though t of as sequen t ia lly unfolding eac h strip b et w een consecutiv e lay ers to a quadrant of the plane, see the left side of Fig. 3). It is clear that the v ertices on ℓ form a free collinear set: if the neighbor ing v ertices of t 2 i − 1 and t 2 i +1 are displaced, then p 2 i is to b e shifted appropriately . In the redra wing π ′′ the roles of o dd and ev en indices are inte rc hanged, that is, t 2 , t 4 , t 6 , . . . ar e put on ℓ a nd t 1 , t 3 , t 5 , . . . on p erp endiculars (see the right side of Fig. 3). It remains to observ e that at least one of t he inequalities ˜ v ( G, π ′ ) ≥ n/ 2 and ˜ v ( G, π ′′ ) ≥ n/ 2 m ust b e true.  10 ALEXANDER RA VSKY and OLEG VERBITSKY t 1 t 2 t 3 t 4 t 5 t 1 t 2 t 3 t 4 t 5 Figure 3. P ro of of L emma 4.3: t wo r edrawings of the gra ph fr o m Fig. 2. Com bining Lemmas 4.2 and 4.3 with Theorem 4.1, w e obta in anot her pro o f for the followin g result. Corollary 4.4 (Goao c et al. [10]) . F or a n y outerplanar gr aph G with n vertic es, we have fix ( G ) ≥ p n/ 2 . In fact, Theorem 4.1 has a m uc h broader range of application. The class of 2-tr e es is defined recursiv ely as follo ws: • the graph consisting of tw o adjacen t v ertices is a 2-tree; • if G is a 2-tree and H is obtained from G b y adding a new v ertex and connecting it to t w o adjacen t v ertices of G , then H is a 2-tree. A graph is a p artial 2 - tr e e if it is a subgraph of a 2-tree. It is w ell know n that the class of partial 2-trees coincides with the class of g raphs with treewidth at most 2. Any outerplanar gra ph is a partial 2-tree, and the same holds for series-parallel graphs (the lat t er class is sometimes defined so tha t it coincides with the class o f partial 2-trees). Note that not all 2-trees are trac k dra wable (for example, the graph consisting of three triangles that share one edge). Theorem 4.5. If G is a p artial 2-tr e e wi th n vertic es, then ˜ v ( G ) > n/ 30 . Corollary 4.6. F or any p artial 2-tr e e G with n ve rtic es we have fix ( G ) ≥ p n/ 30 . 5. The proof of Theorem 4.5 5.1. Outline of the pro of. G iv en tw o v ertex-disjoint 2 -trees, we can add three edges so that the g raph obtained is a 2-tree. It readily follo ws that any partia l 2-tree G is a spanning subgraph of some 2-tree H (that is, V G = V H ). The following lemma, therefore, sho ws that it is enough to prov e Theorem 4.5 for 2- t r ees. Lemma 5.1. If G is a sp ann i n g s ub gr aph of a planar gr aph H , then ˜ v ( G ) ≥ ˜ v ( H ) . Pr o of. If X is a free set of collinear v ertices in a dra wing of H , then X sta ys free in the induced drawing of G .  ON COLLIN EAR SETS I N STRA IGHT LINE DR A WI NGS 11 Figure 4. A 2-t r ee and its folded drawing. Th us, from no w on w e supp o se that G is a 2-tree. W e will consider plain dra wings of G ha ving a sp ecial shap e. Specifically , w e call a dra wing folde d if for any t wo triangles t hat share an edge, one con tains the o t her. Th us, all the triang les sharing an edge form a containmen t c ha in, see Fig . 4. F olded drawings can b e obtained b y the follow ing recursiv e pro cedure. Suppose that G is obta ined from a 2- tree G ′ b y attac hing a new v ertex v to an edge e of G ′ . Then, once G ′ is dra wn, w e put v inside that triangular f ace of the current drawing of G ′ whose b oundary con ta ins e . The pro cedure can b e implemen ted in man y wa ys giving differen t o utputs; all o f them are called folded dra wings. In fact, we will use a pa r a llel v ersion of the pro cedure pro ducing folded dra wing, whic h is in Section 5.2 called Fold . The parallelization will mu c h f acilitate pro ving b y induction that ev ery set of collinear v ertices in a folded drawing is free. The latter fact will reduce our ta sk t o constructing fo lded dra wings with linear n umber of collinear vertice s. In Sections 5.3 and 5.4 w e will analyze tw o sp ecialized v ersions of F old , called Fold 1 and F old 2 , and sho w tha t, for each G , at least one of them succeeds in pro ducing a large enough collinear set. Some definitions. The v ertex set and the edge set of a graph G will b e denoted, resp ectiv ely , b y V ( G ) and E ( G ). Let G b e a 2-tree with n v ertices. Th us, G is obtained in n − 2 steps fro m the single-edge gra ph. Since exactly one new triangle app ears in eac h step, G has n − 2 tr iangles. Denote the set of all tria ng les in G b y △ ( G ). With G w e asso ciate a g r aph T G suc h that V ( T G ) = E ( G ) ∪ △ ( G ) and E ( T G ) consists of all pairs { u v , uv w } where uv w ∈ △ ( G ). A simple inductiv e argumen t sho ws that T G is a tree. 5.2. F olded drawings. A subfold o f a geometric triangle AB C is a crossing-free geometric graph consisting of three triangles A ′ B C , AB ′ C , a nd AB C ′ suc h that all of them are inside AB C , see the first part of Fig. 5 . A 2-tr ee G with a designated triangle abc ∈ △ ( G ) will b e called r o ote d (at the triangle abc ) . Since abc has degree 3 in T G , remo v al of abc from T G splits this tree in to thr ee subtrees. Let T ab denote the subtree containing the ve r t ex ab ∈ E ( G ) . Note that T ab = T G ′ for a certain 2-tree G ′ , a subgraph of G . W e will denote G ′ b y G ab,c . The 2-trees G bc,a and G ac,b are defined symmetrically . No w, giv en a 2- t ree G , w e de fine the class of folde d dr awings of G . Such a dra wing is obtained b y draw ing an arbitrary triangle abc ∈ △ ( G ) a s a geometric triangle AB C a nd then recursiv ely dra wing each of G ab,c , G ac,b , and G bc,a within smaller geometric tria ngles fo r ming a subfold of AB C . More sp ecifically , giv en a 12 ALEXANDER RA VSKY and OLEG VERBITSKY A B C C ′ B ′ A ′ A B C Figure 5. Subfolds of depth 1 and 2. 2-tree G with r o ot abc ∈ △ ( G ) and a geometric triangle AB C , consider the following pro cedure. F old ( G, abc, AB C ) • dra w abc as AB C ; • c ho ose three p oints A ′ , B ′ , and C ′ inside AB C specifying a subfold o f this triangle; • designate ro ots abc ′ , ab ′ c , and a ′ bc in 2- t rees G ab,c , G ac,b , and G bc,a resp ec- tiv ely (if an y of these 2-trees is empty , this branch of the pro cedure termi- nates); • in v o ke F old ( G ab,c , abc ′ , AB C ′ ), F old ( G ac,b , ab ′ c, AB ′ C ), and Fold ( G bc,a , a ′ bc, A ′ B C ) (note that the three subroutines can b e executed indep enden tly in pa r a llel). It is clear that F old ( G, abc, AB C ) produces a drawing of G with outer face AB C . A dra wing of G is called fo lde d if it can b e obtained as the output of F old ( G, abc, AB C ). Note that the describ ed pro cedure is nondeterministic as w e ha ve a lot of freedom in choosing p oin t s A ′ , B ′ , C ′ and we can also hav e a choice of v ertices a ′ , b ′ , c ′ . W e no w intro duce some notions allowing us to sp ecify an y particular computational path of Fold . An y subfold of a geometric triangle AB C specified b y p oin ts A ′ , B ′ , C ′ will b e called a depth-1 subfold o f AB C , and the triang les A ′ B C , AB ′ C , and AB C ′ will b e referred to as its triangular faces (the outer face, which is also tria ngular, is not tak en in to accoun t ) . A depth- ( i + 1 ) subfold of AB C is obtained from an y depth- i subfold b y subfolding eac h of its 3 i triangular faces (thereb y increasing t he n umber of triangular faces to 3 i +1 ), see Fig. 5. Giv en a 2-t ree G , let R b e a tree with V ( R ) = △ ( G ) ro oted a t abc ∈ △ ( G ). Sp ecification of a ro ot determines the standard paren t- child relation on △ ( G ). W e call R a subr o oting tr e e for G if ev ery xy z ∈ △ ( G ) has at most three c hildren and those are of t he form x ′ y z , xy ′ z , xy z ′ for some x ′ , y ′ , z ′ ∈ V ( G ). Note that remo v al of the ro o t abc from R splits it into three subtrees R ab , R bc , and R ac (some ma y b e empt y) suc h that V ( R ab ) = △ ( G ab,c ), V ( R bc ) = △ ( G bc,a ), and V ( R ac ) = △ ( G ac,b ). A similar fact holds t r ue for a n y ro oted subtree of R (a ro o ted ON COLLIN EAR SETS I N STRA IGHT LINE DR A WI NGS 13 subtree is formed by its ro ot xy z and all descendan ts of xy z ). This observ ation mak es eviden t t ha t eac h computational pat h of F old ( G, abc, AB C ) is determined b y a subro oting tree R ro oted at abc and a depth- k subfold S of t he triang le AB C (where k is equal t o the height of R ) : whenev er the subroutine F old ( H , xy z , X Y Z ) is in vok ed for some 2-subtree H of G , w e choo se the v ertices x ′ , y ′ , z ′ and the p oints X ′ , Y ′ , Z ′ so that x ′ y z , xy ′ z , xy z ′ are the c hildren o f xy z in R , and the g eometric triangles X ′ Y Z , X Y ′ Z , X Y Z ′ form the subfold of X Y Z in S (more precisely , in the frag ment of S of the corresp onding depth). W e will denote this path of the pro cedure Fo ld ( G, abc, AB C ) by Fold ( R, S ). Our goal is no w to establish the following fact. Lemma 5.2. Every c ol line ar set of vertic es in a folde d dr awing is fr e e. W e w ill deriv e Lemm a 5.2 from an elemen tary geometric fact stated b elo w as Lemma 5.3, but first we mak e a few useful o bserv ations and introduce some tec hnical notions. Note tha t a depth- k subfold is, in a sense, unique and rigid. More precisely , an y t wo subfolds of the same depth are equiv alent up to homeomorphisms of the plane (as usually , it is supp o sed that a homeomorphism tak es v ertices of geometric graphs to v ertices and edges to edges). Moreo ve r, if S is a depth- k subfold of a triangle AB C and a homeomorphism h ta k es S on t o itself and fixes the ve r t ices A , B , and C (i.e., h ( A ) = A etc.), then h fixes ev ery v ertex of S . It follows that, once w e iden tified the v ertices of the triangle by the labels A , B , and C , eac h other v ertex X of S can b e identifie d by a c anoni c al lab el c ( X ). F ormally , a c anonic al lab eling of a depth- k subfold S is an injectiv e map c : V ( S ) → L , where a set of lab els L con tains the lab els A , B , and C that are assigned to the vertice s of the outer triangle, suc h that any homeomorphism b etw een t w o subfolds resp ecting the lab els A, B , C must resp ect the la b els of all ve r tices. A canonical lab eling can b e defin ed explicitly in man y , essen tially equiv alen t, w ay s. T o b e sp ecific, w e can fix the following definition. Let S b e a depth- k subfold of a triangle AB C obtained b y a sequence AB C = S 0 , S 1 , . . . , S k = S , where S i +1 is obtained from S i b y subfolding eac h inner triang ular face. Note that the sequence S 1 , . . . , S k is reconstructible f rom S . F or example, the vertex C ′ 6 = C that is a djacen t t o A and B in S 1 is determined b y t he condition that AB C ′ is the second largest triangle in the inclusion-c hain of tria ngles in S con taining AB . No w, w e inductiv ely extend the canonical lab eling c from V ( S 0 ) to V ( S ) as follow s: if X Y Z is a triangula r face of S i +1 and X ∈ V ( S i +1 ) \ V ( S i ) (hence Y , Z ∈ V ( S i )), then c ( X ) = ( i + 1 , c ( Y Z )). F urthermore, let ℓ b e a line suc h that no edge of S lies on it. By S ∩ ℓ we will denote the set of common p oints of S and ℓ , consisting of the vertices of S lying on ℓ and the p oin ts of in t ersection of ℓ with edges of S . In addition to the canonical lab eling of t he ve rtices o f S , w e lab el eac h in tersection p oin t of ℓ with an edge Y Z of S b y c ( Y Z ). This a llo ws us to define the interse ction p attern o f S and ℓ to b e the sequence of the lab els of all p oin ts in S ∩ ℓ as they app ear along ℓ (thus , this sequence is defined up to rev ersal). 14 ALEXANDER RA VSKY and OLEG VERBITSKY Figure 6. Base case in the pro of of Lemma 5.3 Giv en t w o triangles AB C and A ′ B ′ C ′ , we will iden tify the matc hing vertex names, that is, A and A ′ , B and B ′ , and C and C ′ . Lemma 5.3. Supp ose that a line ℓ interse cts a triangle AB C in at le ast 2 p oints and the same is true for a line ℓ ′ and a triangle A ′ B ′ C ′ . Mor e over, let AB C ∩ ℓ and A ′ B ′ C ′ ∩ ℓ ′ have the same interse ction p attern (for example, if ℓ p asses thr ough A and cr osses B C , then ℓ ′ p asses thr ough A ′ and cr osses B ′ C ′ ). Co n sider an arbitr ary depth- k subfold S of AB C and an arbitr ary set of p oints P within A ′ B ′ C ′ such that A ′ B ′ C ′ ∩ ℓ ′ ⊂ P ⊂ ℓ ′ and | P | = | S ∩ ℓ | . Then ther e exists a depth- k subfold S ′ of A ′ B ′ C ′ such that S ′ ∩ ℓ ′ = P and, m o r e ove r, S ′ ∩ ℓ ′ has the sa me interse ction p attern as S ∩ ℓ . Pr o of. W e pro ceed b y induction on k . The base step o f k = 1 is a trivial geometric graph, see Fig. 6. Supp ose that k > 1. Then S is obtained from a depth-1 subfold S 1 of AB C by depth-( k − 1) subfolding eac h of its three triangular faces. Lab el the p oin ts of P according to the interse ctio n pattern of S ∩ ℓ . Let P ◦ ⊆ P consist o f the p oin ts whose lab els app ear in S 1 ∩ ℓ . Since the lemma is t r ue in the base case, there is a subfold S ′ 1 of A ′ B ′ C ′ suc h that S ′ 1 ∩ ℓ ′ = P ◦ and t he inters ection pattern of S ′ 1 ∩ ℓ ′ agrees with the lab eling of P ◦ . Let F 1 , F 2 , F 3 b e the triangular faces of S 1 and F ′ 1 , F ′ 2 , F ′ 3 the corresp onding triangula r fa ces of S ′ 1 . F or eac h i = 1 , 2 , 3, let P i b e the segmen t of P inside F ′ i (it ma y b e empt y for some i ). There remains to apply the induction h yp o thesis to the lines ℓ and ℓ ′ , the triangles b ounding F i and F ′ i , the depth-( k − 1) subfold of F i induced b y S , and the p o in t set P i , for eac h i = 1 , 2 , 3.  Pr o of of L emma 5.2. Given a folded drawing σ a nd a line ℓ , w e ha ve to sho w that V ( σ ) ∩ ℓ is free. Let AB C b e the b oundary of the outer face of σ . It suffices to consider the case when σ is a depth- k subfold of AB C , for some k (completing σ , if necessary , w e will pro ve ev en a stronger fact). Using t he notio n o f a canonical lab eling, our task can b e stated a s follow s: Given a n arbitrary set P ′ of | V ( σ ) ∩ ℓ | p oin ts o n a line ℓ ′ , w e hav e to find a depth- k subfold σ ′ of some triangle A ′ B ′ C ′ suc h that V ( σ ′ ) ∩ ℓ ′ = P ′ and the interse ction pa tterns of σ ∩ ℓ and σ ′ ∩ ℓ ′ agree on V ( σ ) ∩ ℓ and V ( σ ′ ) ∩ ℓ ′ . The solv a bilit y of this ta sk follo ws from Lemma 5 .3.  5.3. F olded drawings with man y c ollinear leaf v ert ices. Let G b e a 2 -tree. W e will call v ∈ V ( G ) a le af vertex if it has degree 2. The t wo edges emanating from v will b e referred to as le af e dge s . A triangle con taining a leaf v ertex w ill b e called a le af triangle . The n um b er of leaf tria ngles in △ ( G ) will b e denoted b y t 1 = t 1 ( G ). Our nearest goa l is to sho w that there is a folded drawing of G with ON COLLIN EAR SETS I N STRA IGHT LINE DR A WI NGS 15 at least 2 3 t 1 collinear leaf vertice s. The graph all whose triangles share an edge can b e excluded from consideration, as it can b e dra wn with all leaf v ertices on a line. This is the only case when a g raph has only leaf triangles. In the sequel w e will, therefore, assume that G contains at least one non-leaf triangle. W e will describ e a sp ecification of the pro cedure Fo ld that, additionally to a 2- t r ee G , tak es on input a line ℓ and pro duces a folded dra wing of G suc h that ℓ crosses eac h non-leaf triangle in tw o edges a nd passes through a leaf v ertex whenev er p ossible. F old 1 ( G, ℓ ) • c ho ose a non- leaf triangle abc ∈ △ ( G ) , a geometric triangle AB C with tw o sides crossed by ℓ , and execute F old ( G, abc, AB C ) o b eying the follo wing additional conditions. Whene v er a subroutine F old ( H , xy z , X Y Z ) is in- v ok ed for a 2-subtree H of G , – if po ssible, t he ro o t xy z for H should be a non-leaf triangle in G . In this case the geometric triangle X Y Z should b e drawn so t hat t wo sides of it are crossed by ℓ ; – if not (i.e., △ ( H ) con tains only leaf t r ia ngles of G ), the leaf v ertex of xy z should b e put on ℓ whenev er this is p ossible. Lemma 5.4. The pr o c e dur e Fold 1 ( G, ℓ ) c an b e run so that it pr o duc e s a folde d dr awing of G with at le ast 2 3 t 1 ( G ) le af vertic es lying on the line ℓ . Pr o of. Given a non- leaf edge e of G , let ω ( e ) denote the n um b er o f leaf triangles con taining e . F or all e w e initially set cr ( e ) = 1, and in the course of execution of F old 1 ( G, ℓ ) we reset this v alue to cr ( e ) = 1 o nce this edge is dra wn and crosses ℓ . Note that, if v is a leaf v ertex in a leaf triangle v uw and cr( uw ) = 1, then F old 1 ( G, ℓ ) puts v on ℓ (lik e an y other leaf v ertex adjacent to u and w ). It follows that the pro cedure puts on ℓ at least X e cr( e ) ω ( e ) (14) leaf v ertices, where the summation go es ov er all non- leaf edges. Thus , w e hav e to sho w that this sum can b e made large. T o this end, consider a randomized v ersion of Fold 1 ( G, ℓ ). Recall that the pro- cedure b egins with dra wing a non-leaf triangle abc ∈ △ ( G ) so that exactly t wo edges of it are crossed b y ℓ . W e can c ho ose the pair of crossed edges in three w a ys and do it at random. Consider now a r ecursiv e step where w e ha ve to dra w a non-leaf triangle xy z whose edge xy is already drawn and the v ertex z still not. If cr( xy ) = 0, then the r ules of Fold 1 ( G, ℓ ) for ce lo cating z so that cr( xz ) = cr ( y z ) = 1. If cr( xy ) = 1, then exactly one of xz and y z can (and mu st) b e crossed b y ℓ . In this case w e mak e this c hoice again a t random. A simple induction (on the distance of e from abc in T G ) sho ws that each non-leaf edge e is crossed b y ℓ with pro babilit y 2 / 3. By linearit y of exp ectation, the mean v a lue of the sum (1 4) is equal to 2 3 X e ω ( e ) = 2 3 t 1 ( G ) . 16 ALEXANDER RA VSKY and OLEG VERBITSKY Figure 7. Chains of length 5 and their folded dra wings. The dashed p olyline sho ws the tra ce of the line ℓ in an unfo lded drawing. Therefore, the randomized ve rsion o f F old 1 ( G, ℓ ) with no nzero probability puts at least 2 3 t 1 leaf v ertices on ℓ . It readily follow s that at least one computational path of F old 1 ( G, ℓ ) pro duces a folding dra wing of G with at least 2 3 t 1 leaf v ertices on ℓ .  5.4. F olded dra wings w it h man y collinear interposed v ert ices. Tw o triangles of a 2-tree G will b e called neig hb ors if they share an edge. Call a triangle T ∈ △ ( G ) linking if it has exactly t wo neighbors and they a re edge-disjoin t. In other w o r ds, T do es not share o ne of its edges with an y other triang le and shares eac h of t he other t wo edges with exactly one tria ng le. The num b er of linking tria ng les in G will b e denoted by t 2 = t 2 ( G ). A maximal sequence o f unequal linking tria ngles T 1 , . . . , T k where T i and T i +1 are neighbors for all i < k will b e called a ch a in of tr ia ngles in G . Here, maximal means that the c hain cannot b e extended to a longer sequence with the same prop erties. Note that different ch ains are disjoint. If k = 1, then t he t wo neigh b o rs of T 1 , T ′ and T ′′ , are non- linking. If k > 1, then T 1 has exactly one non-linking neighbor T ′ , and T k has exactly one non-linking neigh b or T ′′ . In either case, w e sa y that T ′ and T ′′ are c onne cte d b y the c ha in. Lemma 5.5. A 2 -tr e e G with t triangle s has less than t − t 2 chains. Pr o of. Define a graph H on t he set o f all non-linking triangles of G so that t w o triangles are adjacen t in H if they are connected b y a c hain in G . Th us, H has t − t 2 v ertices and exactly a s man y edges as there are c hains in G . It remains to not ice that H is acyclic (otherwise T G w ould con tain a cycle).  The particular case of a 2-tree consisting of a single chain of length t deserv es a sp ecial attention. Note that, if t ≤ 3, suc h a graph unique up to isomorphism. F urthermore, there are tw o isomorphism ty p es if t = 4 and fo ur isomor phism types if t = 5. F or the latter case, all p ossibilities are show n in Fig. 7, where w e see also folded drawings of the four graphs that hav e some useful prop erties, as stated b elow. Lemma 5.6. L et G b e a 2-tr e e c onsisting of a s ingle chain of length 5. Given a line ℓ , ther e is a folde d dr awing of G wher e ℓ p asses thr ough o n e vertex a nd cr osses two e dges in b oth end triangles. Call a ve rtex interp ose d if it b elongs only to linking tr ia ngles. W e now describe a sp ecification of the pro cedure Fold that aims to pro duce a folded drawing of a 2-tree G with many interposed v ertices on a line ℓ . ON COLLIN EAR SETS I N STRA IGHT LINE DR A WI NGS 17 T ′ T 1 T 2 T ′ T 1 T 2 Figure 8. T 1 , T 2 , . . . , T 5 is a g roup of linking triangles in a chain. In an y case, T 1 can b e draw n so that ℓ crosses a lso the common edge of T 1 and T 2 . F old 2 ( G, ℓ ) • Ro ot G at a non-leaf and non-linking triangle abc ∈ △ ( G ) . Note that, whatev er subro oting tree is used, eac h chain will app ear in it as a path in the direction from the ro ot abc up w a r ds. Split eac h c hain into gr oups of 5 success ive linking tria ngles, a llowing the last g r oup to b e inc o mplete (i.e., ha ve up to 4 triangles). • Whenev er a subroutine F old ( H , xy z , X Y Z ) is inv o k ed for a 2-subtree H o f G , the fo llo wing rules ha ve to b e ob ey ed. – If p ossible, t he ro ot xy z for H should b e a non-leaf triang le in G . – If xy z is neither a leaf nor a linking tr ia ngle in G , then ℓ should cross t w o sides of X Y Z . The same applies t o an y linking triangle in an incomplete group. – F o r eac h complete group of linking triangles, its in tersection patt ern with ℓ should ha v e t he prop erties claimed b y Lemma 5.6. This is al- w ay s p ossible, irresp ectiv ely of the crossing pattern of the tr iangle x ′ y ′ z ′ preceding this group in the subro o t ing tree, see Fig. 8 (note that x ′ y ′ z ′ cannot b e a leaf triangle a nd, hence, is crossed b y ℓ in t wo edges). Lemma 5.7. L et G b e a 2-tr e e with n ve rtic es and t 2 linking triangles. The pr o c e dur e F old 2 ( G, ℓ ) pr o duc es a folde d dr awing of G with mor e than t 2 − 4 5 n interp ose d vertic es lying on the lin e ℓ . Pr o of. Denote the n um b er of interposed v ertices on ℓ b y l . The rules of Fo ld 2 for dra wing chains ensure that l is equal to the n umber of complete groups of linking triangles in G . Let c denote the n umber of c hains in G , which is the trivial upp er b ound for the n umber of incomplete groups. W e, therefore, hav e t 2 ≤ 5 l + 4 c. By Lemma 5.5, we a lso hav e c ≤ n − t 2 − 3 . It f ollo ws that 5 l ≥ 5 t 2 − 4 n + 12, yielding the desired b ound.  5.5. The r est of t he pro of. Notice that the rules of Fold 1 and Fold 2 are coher- en t and w e can consider a h ybrid pro cedure F old 1+2 ( G, ℓ ). This pro cedure aims at lo cating on ℓ as many leaf v ertices a s F old 1 ( G, ℓ ) do es and as many in terp osed v ertices as Fo ld 2 ( G, ℓ ) do es. The la tter g oal is ac hieve d by adopting t he instruc- tions of F old 2 for drawin g c hains. The f o rmer goal is ac hieve d, like F old 1 , b y randomization. In order to ensure tha t, with nonzero probabilit y , at least 2 3 t 1 ( G ) v ertices are put on ℓ , w e need to fulfill the follo wing condition: 18 ALEXANDER RA VSKY and OLEG VERBITSKY T 4 T 5 T ′′ with probability 1 / 3 T 4 T 5 T ′′ with probability 2 / 3 Figure 9. On leavin g the la st complete gro up in a chain. (*) if T is neither a leaf tria ngle nor a linking triangle in a complete g roup, then eac h edge of T is crossed by ℓ with probability 2 3 . The exceptional treatment of linking triang les do es not decrease the c hances of an y leaf v ertex to b e put on ℓ . Indeed, either a linking triangle T has no leaf t riangle in the neigh b orho o d or T is the end triangle in a ch ain neighboring with a leaf tria ngle T ′ . In t he latter case t he common edge of T and T ′ can be crossed, see F ig . 7 (hence, the leaf v ertex of T ′ will b e put on ℓ ). Ho we v er, some car e is needed to fulfill Condition (*) on leaving the last complete group T 1 , T 2 , T 3 , T 4 , T 5 in a c ha in. Notice that in each of the f o ur cases show n in Fig. 7 w e ha ve tw o choices fo r an edge o f the end triangle T that will b e crossed by ℓ . W e mak e t his choice at random, with probability distribution sho wn in Fig. 9. This ensures (*) fo r the next neighbor T ′′ of T 5 . Th us, b oth Lemmas 5.4 and 5.7 a pply as w ell to Fold 1+2 and, therefore, this pro cedure enables lo cating more than 2 3 t 1 + t 2 − 4 5 n (15) v ertices of G on ℓ . Our further analysis is based on the follow ing lemma. Lemma 5.8. If G is a 2-tr e e with n ≥ 3 vertic es, t 1 le af triangles , and t 2 linking triangles, then 4 t 1 + t 2 > n . Pr o of. Denote the num b er of all triangles in G by t and recall that t = n − 2. Th us, w e hav e to prov e that 4 t 1 + t 2 ≥ t + 3 . (16) W e pro ceed by induction on t . If t = 1 , that is, G is a tr iangle, the inequality is true. Suppo se that t ≥ 2. If G has a linking tria ng le uv w , where the edge uv is not shared with any other triangle, let G ′ b e obtained from G by con traction o f u v . F or the corresp onding parameters o f G ′ , w e hav e t ′ = t − 1, t ′ 2 = t 2 − 1 , and t ′ 1 = t 1 . Inequalit y (16) readily follo ws from the induction assumption. Assume no w that t 2 = 0. Consider an arbitrary leaf triang le u v w , with w b eing a leaf triangle. Our analysis is split into a f ew cases, see Fig . 1 0. Case 1 : uv w ha s a single neigh b or uv z . Since (16) is true if G is the diamond graph, w e supp ose that n ≥ 5. Sub c ase 1-a: uv w sha r e s b oth e d ges uz and v z with other triangles. Let G ′ = G − w . W e hav e t ′ = t − 1 and t ′ 1 = t 1 − 1. Note that the triangle uv z can b ecome ON COLLIN EAR SETS I N STRA IGHT LINE DR A WI NGS 19 z v u w Case 1-a z v u w Case 1-b v u w Case 2 Figure 10. Pro of o f Lemma 5.8 (illustrative fragments of G ). linking; then w e will hav e t ′ 2 = 1. The induction assumption applied to G ′ giv es us 4 t 1 ≥ t + 5. This inequality is ev en stronger than (16), as t 2 = 0. Sub c ase 1-b: uv w do es not shar e one o f its e dges , say uz , with any other triangle. Let G ′ = G \ { u, w } , so that t ′ = t − 2. Since the tr ia ngle uv z is not linking, it shares the edge z v with k ≥ 2 neigh b ors. This implies that t ′ 1 = t 1 − 1. If k = 2, the tw o triangles sharing z v with uv z can b ecome linking in G ′ , and w e will ha v e t ′ 2 = 2. The induction assumption applied to G ′ giv es us 4 t 1 ≥ t + 3, which is the same as (16) b ecause t 2 = 0. Case 2: uv w has k ≥ 2 neighb ors. Let G ′ = G − w . Clearly , t ′ = t − 1 and t ′ 1 = t 1 − 1. If k > 2, we hav e t ′ 2 = 0. If k = 2, it can happ en that the t wo tr ia ngles sharing uv with uv w b ecome linking in G ′ raising the v alue of t ′ 2 to 2. Again, (16) follo ws from the induction assumption a pplied to G ′ .  An example in Fig. 11 sho ws t ha t the factor of 4 in the b ound (16) cannot b e impro ve d. T urning back to the pro of of Theorem 4.5, supp ose first that t 2 ≤ 4 5 n . In this case the b ound (15) has no adv an tage up on the p erfor mance of Fold 1 ( G, ℓ ), that puts at least 2 3 t 1 v ertices on ℓ . By Lemma 5.8, w e then hav e t 1 > n − t 2 4 ≥ n 20 , and hence ℓ passes trough more than n/ 30 vertice s. If t 2 > 4 5 n , the pro cedure Fold 1+2 is preferable and yields 2 3 t 1 + t 2 − 4 5 n > n − t 2 6 + t 2 − 4 5 n = 25 t 2 − 1 9 n 30 > n 30 . collinear v ertices. T he pro of is complete. Figure 11. A 2-tr ee with t = 4 k + 2 triang les, f or whic h t 1 = k + 2 and t 2 = 0 (the parameter k can tak e an arbitra r y v alue). 20 ALEXANDER RA VSKY and OLEG VERBITSKY 6. Questions and comments 1. How far or close are pa r a meters ˜ v ( G ) and ¯ v ( G ) ? It seems that a priori w e ev en cannot exclude equalit y . T o clarify t his question, it would b e helpful to (dis)prov e that ev ery collinear set in any straight line dra wing is fr ee. 2. W e constructed examples o f graphs with ˜ v ( G ) ≤ ¯ v ( G ) ≤ O ( n σ + ǫ ) for a graph- theoretic constan t σ , for whic h it is kno wn that 0 . 753 < σ < 0 . 99. Are there graphs with ¯ v ( G ) = O ( √ n )? If so, this could be considered a strengthening of the examples of graphs with fix ( G ) = O ( √ n ) give n in [3 , 10, 14]. Are there graphs with, a t least, ˜ v ( G ) = O ( √ n )? If not, by Theorem 4.1 this w ould lead to an impro veme n t of Bose et al.’s b ound (2 ). 3. By Theorem 4.5, we hav e ˜ v ( G ) ≥ n/ 30 for an y gra ph G with tree-width no more than 2. One can a lso show that for Ha lin graphs, whose tree-width can attain 3, we ha ve ˜ v ( G ) ≥ n/ 2. F or which other classes of graphs do we ha ve ˜ v ( G ) = Ω( n ) or, at least, ¯ v ( G ) = Ω( n )? In particular, is ¯ v ( G ) linear for 2-outerplanar graphs? These graphs ha v e tr ee-width at most 5, and one can extend this question to planar graphs with tree-width b ounded by a small constant t . Corollary 3.5 gives a negativ e answ er if t is suffic iently large. F urthermore, what ab out planar g r aphs with b ounded v ertex degrees? Note that the graphs constructed in the pro of of Theorem 3.3 ha v e vertice s with degree more than n δ for some δ > 0. 4. In a recen t pap er [4], Cano, T´ oth, and Urrutia improv e the upp er b ound (4) to a b ound of O ( n 1 / (3 − σ )+ ǫ ). Similarly to our pro o f of Theorem 3.3, their construction also uses iterative r efinemen t of faces of a planar triangulation. It fo llows that our lo we r b ound of O ( √ n ) in Corollary 4.6 cannot b e extended to an y class of planar graphs with b ounded tree-width, ev en to planar graphs of tree-width 8. If suc h extension is p ossible for tree-width 3,4,. . . is a natural o p en problem. 5. Whether or not fit ( G ) = ¯ v ( G ) is an in triguing question. Similarly to (1 3 ), one can prov e that fit ( G ) = fit − ( G ), where fit − ( G ) = min X fit X ( G ) with the minimiza- tion o ve r collinear X . Th us, the question is actually whether or not fit X ( G ) has the same v alue for all collinear X . A similar question for fix X ( G ) is also op en; it is p osed in [14 , Problem 6.5]. 6. It is also natural to consider fi t ∨ ( G ) = min X fit X ( G ), where the minimization go es o ve r all X in general p o sition. Can o ne e xtend our upp er b o und fi t ( G ) = O ( n 0 . 99 ) to show that fit ∨ ( G ) = o ( n ) for infinitely many G ? 7. By sligh tly mo difying the pro of of Lemma 3.1.2, one can show t hat c ( G ∗ k ) ≤ ( c ( G ∗ 1 ) − 1) k − 1 c ( G ∗ 1 ). It fo llo ws that, for an y cubic p o lyhedral graph H , there ex- ists a seque nce of cubic p olyhedral g raph H 1 , H 2 , . . . suc h that lim k →∞ log c ( H k ) log v ( H k ) ≤ log( c ( H ) − 1) log( v ( H ) − 1) < log c ( H ) log v ( H ) . This readily implies t wo prop erties of the shortness expo nen t for cubic po lyhedral graphs, that seem to b e unnoticed so far. First, σ = inf H log c ( H ) log v ( H ) o ve r a ll cubic p olyhedral H . Second, σ cannot b e atta ined b y the fraction log c ( H ) log v ( H ) for an y particular H . It is interes ting if this holds true for other fa milies o f graphs. ON COLLIN EAR SETS I N STRA IGHT LINE DR A WI NGS 21 Reference s [1] M. Bilinski, B . Jackson, J. Ma, X. Y u. Circumference of 3-connected claw-free g raphs and lar ge Eulerian subg r aphs o f 3-edg e-connected g raphs. 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