On the inducibility of oriented graphs on four vertices
aa r X i v : . [ m a t h . C O ] O c t On the inducibility of oriented graphs on four vertices ∗Łukasz Bożyk † Andrzej Grzesik ‡ Bartłomiej Kielak § Abstract
We consider the problem of determining the inducibility (maximum possible asymptoticdensity of induced copies) of oriented graphs on four vertices. For most of the graphs we provethe exact value, while for all the remaining ones we provide very close lower and upper bounds.It occurs that, for some graphs, the structure of extremal constructions maximizing density ofits induced copies is very sophisticated and complex.
Fix a graph H on k vertices. For a graph G , let N ( H, G ) denote the number of induced copiesof H in G , i.e., the number of k -sets of vertices of G that induce graphs isomorphic to H . Define I ( H ) = lim n →∞ max { N ( H, G ) : | V ( G ) | = n } (cid:0) nk (cid:1) . We call I ( H ) the inducibility of a graph H . For H being complete graph or independent set wehave I ( H ) = 1, but in most of the remaining cases the problem of determining I ( H ) is open. Theconcept of inducibility was introduced in 1975 by Pippenger and Golumbic [17], who observed that I ( H ) ≥ k ! k k − k for any graph H on k vertices, and conjectured the following: Conjecture 1. If H is a cycle of length k ≥ , then I ( H ) = k ! k k − k . Since then, many results on inducibility were obtained, including the inducibility of completebipartite graphs [4, 5], multipartite graphs [3], and blow-ups of graphs [10]. Recently, also weakerbounds in Conjecture 1 were obtained [11, 16] and the conjecture was proved in the case k = 5by Balogh, Hu, Lidický, and Pfender [1]. Inducibility was determined exactly for all graphs onfour vertices [8, 12] with a single exception of the four-vertex path P , for which we still do noteven have a conjecture about what should be the extremal construction. The best lower boundis ≈ . ≈ . flagmatic software [19].Inducibility can be defined in the same way also for other discrete structures, including hyper-graphs, multigraphs, directed graphs, and oriented graphs. In our work we focus on the latter.Recall that by an oriented graph we mean a graph with directed edges which has no loops, nomultiple edges, and no bidirected edges (2-cycles), i.e., a simple graph with an orientation of theedges.In general, little is known about the inducibility of oriented graphs. Huang [15] determinedthe inducibility of oriented stars −→ S k for all k ≥ −→ K s,t for2 ≤ s ≤ t . Recently, Hu et al. [14] determined the inducibility of all other orientations of stars onat least 7 vertices. There is also an analogue of Conjecture 1 for oriented graphs: ∗ This work was supported by the National Science Centre grant 2016/21/D/ST1/00998. † Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa,Poland. E-mail: [email protected] . ‡ Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland.E-mail:
[email protected] . § Faculty of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland.E-mail: [email protected] . onjecture 2. If H is a directed cycle of length k ≥ , then I ( H ) = k ! k k − k . The case k = 4 was proved by Hu, Lidický, Pfender, and Volec [13], who also determinedinducibility of all other orientations of C . Note that the case k = 5 follows from the result onundirected cycles [1].The inducibility of all oriented graphs on 3 vertices is known. For directed triangle it is 1 / / √ − / I ( H ) = I ( ←− H ), where ←− H is the graph obtained from H byreversing all arcs, the number of non-isomorphic cases to consider can be reduced to 30.In Section 3 we present upper bounds and constructions providing lower bounds for all orientedgraph on four vertices. Few of them follow from the results on the inducibility of undirectedgraphs [8], whereas star [15], tournaments [6] and orientations of C [13] were proven earlier.In the remaining cases, the upper bounds were obtained mostly using flagmatic software [19]based on the flag algebra method of Razborov [18]. In several cases, presented constructionsgive lower bounds that are matching the upper bounds, while in the remaining ones, when theconstruction is complex, the lower bounds differ by at most 0 .
004 in one case, and by at most0 .
001 in all the other cases. This indicates that the constructions might be correct and the appliedflag algebras computations were not enough to obtain the matching upper bounds. All of the resultsare summarized in Table 1. Whenever the constant is irrational, it is defined in the appropriatesubsection of Section 3. Description of the used notation and explanations of the pictogramsapplied to illustrate the constructions are contained in Section 2.It is worth mentioning that the obtained results indicate the structure of constructions givingthe inducibility to be far richer than the intuition suggests. Yuster [20] and independently Fox,Huang, and Lee [9] proved that for almost all graphs H , the inducibility is attained by the iteratedblow-up of H . For small graphs the situation is different. Out of 28 non-trivial non-isomorphicgraphs considered here, only 2 of them have this property, whereas such constructions as those inSubsections 3.4, 3.20, or 3.23 show that the inducibility can be attained by very sophisticated andcomplex structures. For a vertex v ∈ G , denote by d + ( v ), d − ( v ), and d ′ ( v ) := | G | − − d + ( v ) − d − ( v ) its outdegree,indegree, and nondegree, respectively.By A n , T n , and T reg n we denote, respectively, the empty graph, the transitive tournament, andany regular tournament on n vertices. For a graph G , the oriented graph G rand denotes a graphobtained from G by orienting every edge independently at random.By G ⊙ H we denote the lexicographic product of G and H . We also write G ⊙ n to indicatea graph G ⊙ . . . ⊙ G , an iterated blow-up of G .For any graphs G and H , G ⊔ H denotes the graph obtained by taking disjoint union of graphs G and H . If G and H are oriented graphs, then we write G p → H to denote a graph constructedfrom G ⊔ H by joining each vertex of G with each vertex of H by an arc independently at randomwith probability p . For p = 1, we just write G → H .For any a ∈ [0 , ], we define a circular graph S ( a ) with vertex set [0 ,
1) and edge from x to x + α mod 1 for each x ∈ [0 ,
1) and each α ∈ [0 , a ).While depicting constructions corresponding to obtained lower bounds, we use the followingconventions. First of all, the illustrations show the limit structure of each construction instead ofthe finite graphs forming a sequence giving the lower bound. This allows to see the structure of thissequence without caring about getting integer sizes of particular blobs. To each blob is assigned areal number from the interval (0 ,
1) which corresponds to the fraction of vertices present in thatblob in the limit (in a corresponding n -vertex graph, a blob of size α is assumed to have roughly2d H , ←− H Upper bound Lower boundId H , ←− H Upper bound Approximation Value Construction1 1 1 1 A n / A ⊙ T n / T n ⊔ T n ≈ c G ≈ −→ C ⊙ n ⊔ −→ C ⊙ n ≈ / −→ C ⊙ n ≈ c G / A n / → A n c c G
10 0.2025 0.2025 81 / G
11 0.125 0.125 1 / T reg n ⊔ T reg n
12 0.5 0.5 1 / T n ⊔ T n
13 2/21 0.095238 2 / −→ C ⊙ n
14 0.1875 0.1875 3 / K rand n,n c c G
16 0.375 0.375 3 / −−−→ K n,n ≈ H ⊙ n ≈ c G ≈ c G ≈ c G c c S ( √ )22 ≈ c G ≈ c G
24 0.375 0.375 3 / T n → A n
25 4 / / −→ C ⊙ A n ≈ c G ≈ S (4 / c c G
29 0.5 0.5 1 / S (1 / T n Table 1: Summary of bounds on inducibility of graphs on four vertices.3 n vertices). If no blob sizes occur, all parts in the picture are meant to have equal sizes. Everyblob forming an independent set is depicted as a white (empty) circle, while a blob forming anarbitrary tournament is depicted with a gray (shaded) circle. When some other structure appearsinside the blob, then there is a letter indicating the structure. In particular, T for the transitivetournament and R for the random tournament. Iterated constructions are marked by a dot insidethe circle—this means that the blob consists of a copy of the entire construction. We also provide aspecial pictogram for a circular graph S ( α ). All the above pictograms are summarized in Figure 1. T R S α (a) (b) (c) (d) (e) (f) (g)Figure 1: (a) Anticlique. (b) Arbitrary tournament. (c) Transitive tournament. (d) Random tour-nament. (e) Iterative structure. (f) Blob with structure S . (g) Circular graph with parameter α . In the following subsections, for each oriented graph on 4 vertices we present the proven upper andlower bound on its inducibility and provide schematic picture and description of a constructiongiving the lower bound. If a pair H, −→ H is considered, only a construction for the graph H ispresented; by reversing arcs, one may obtain an analogous construction for the graph −→ H .Each value of the upper bound which is not preceded with an approximation symbol ( ≈ ) isproven exactly and meets the lower bound. Whenever we use flagmatic software [19] we indicateon how many vertices it is performed. Unless it is possible to make some reduction by forbiddingcertain structures in the extremal construction, the computations are performed using graphs onat most 6 vertices. It is possible to increase this number, which will result in slight improvementof the upper bounds, but will cause much longer running time of the program.In almost all cases with sharp bounds, the standard application of the flag algebras semidefinitemethod is insufficient, due to rounding errors made in the rationalization process. In order toovercome this difficulty, we applied the method described in [2] of finding eigenvectors of zeroeigenvalue of the numerically obtained semidefinite matrix. For each calculation we publish theapplied flagmatic code with all commands and added eigenvectors, as well as the certificateuseful for verification of the obtained bound. All the codes and certificates are available with theelectronic preprint of this manuscript on arxiv.org . Graph:Upper bound: 1Lower bound: 1Construction:Anticlique.
Graph:Upper bound: 72/125 4s each weak component of this graph is a transitive tournament, the inducibility of this graphis equal to the inducibility of the non-oriented graph K ⊔ A (an edge on four vertices), whichwas explicitly determined in [8].Lower bound: 72/125Construction:Balanced union of arbitrary five tournaments. Graph:Upper bound: 3/8As each weak component of this graph is a transitive tournament, the inducibility of this graphis equal to the inducibility of the non-oriented graph K ⊔ K , which is equal to the inducibility ofits complement K , = C , whose value is known [5].Lower bound: 3/8Construction:Balanced union of arbitrary two tournaments. Graph:Upper bound (by flagmatic on 6 vertices): ≈ ≈ x i = y i for all i ≥ x ′ i = y ′ i for all i ≥ x ′ = x ′ i = 0 for i ≥
6, and ( x i ) i ≥ is a geometric series.Construction: y ′ y ′ y ′ y ′ x x x · · ·· · ·· · · x ′ x ′ x ′ x ′ y y y · · ·· · · · · · x ′ + y ′ + P i ≥ x i + y i + x ′ i + y ′ i = 1 Construction G is the following: split vertices into parts X i and Y i , i ≥
0. Consider also partitions X = S ∞ i =0 X ′ i and Y = S ∞ i =0 Y ′ i . Add all edges from X j to X i and from Y j to Y i for 0 ≤ i < j .Finally, add all edges from X i to S ij =1 Y ′ i and from Y i to S ij =1 X ′ i for i ≥ .5 Graph 5 Graph:Upper bound (by flagmatic on 6 vertices): ≈ / ≈ . −→ C ⊙ −→ C ⊙ Balanced union of two iterated balanced blow-ups of −→ C . Graph:Upper bound (by flagmatic on 6 vertices): ≈ / ≈ . −→ C . Graph:Upper bound (by flagmatic on 6 vertices): ≈ c ≈ . c = max a,b,c,d ∈ [0 , a +2 b +2 c + d =1 a + b ) c + 2 abd ( b + 2 c ))1 − a − b − c − d . Construction: a c c ab d b
Construction G is a weighted iterated blow-up of a graph on 7 vertices. More specifically, let2 a + 2 b + 2 c + d = 1 and split vertices into 7 parts—two of size a , two of size b , two of size c , and6ne of size d . Add edges between appropriate parts to make them complete bipartite and orientthem as shown in the picture. Finally, iterate this process inside each of the seven parts. Notethat the direction of edges between parts of size c is not important, hence many non-isomorphicexamples of graphs with this bound met can be found. Graph:Upper bound (by flagmatic on 6 vertices): 81/512Lower bound: 81/512Construction: / Union of two balanced anticliques with random edges from the first to the second with probability . Graph:Upper bound: c , where c = max x ∈ [0 , ] − x ) x − (1 − x ) = 4 − p √ − q √ − . Proved by Huang [15], also included as an example application of the flagmatic software [19].Lower bound: c ≈ . x − x Split vertices into two parts, A of size x and B of size 1 − x . Then, put all possible edges from A to B and iterate this process inside A . Graph:Upper bound (by flagmatic on 5 vertices): 81/400Lower bound: 81/400Construction:
320 15 3201414 G is the following: split vertices into 5 parts— A of size , B of size , C of size , D of size , and E of size . Put all edges from A to B , from B to C , from C to D , from D to A ,from C to E , and from A to E . In [14] there is also a general probabilistic construction for anyorientation of a star. Graph:Upper bound (by flagmatic on 5 vertices): 1/8Lower bound: 1/8Construction: reg reg
Balanced union of arbitrary two regular tournaments.
Graph:Upper bound: 1/2As each weak component of this graph is a transitive tournament, the inducibility of this graphis equal to the inducibility of the non-oriented graph K ⊔ A (a triangle with an isolated vertex),which was determined in [8].Lower bound: 1/2Construction: T T
Balanced union of two transitive tournaments.
Graph:Upper bound: 2/21 proved in [13].Lower bound: 2/21Construction:Iterated balanced blowup of −→ C . 8 .14 Graph 14 Graph:Upper bound (by flagmatic on 5 vertices): 3/16Lower bound: 3/16Construction:Balanced complete bipartite graph with randomly oriented edges.
Graph:Upper bound: c proved in [13], where c = max x ∈ [0 , ] x (1 − x ) − x = 9 (cid:16) √ − (cid:17) + 6 r (cid:16) √ − (cid:17) . Lower bound: c ≈ . x − x x Construction G is the following: split vertices into three parts, A of size x , B of size 1 − x , and C of size x . Put all possible edges from A to B and from B to C . Iterate this process inside A and inside C . Graph:Upper bound: 3/8It is the inducibility of the non-oriented C , which is K , .Lower bound: 3/8Construction:Complete balanced bipartite graph with edges oriented from the first part to the second.9 .17 Graph 17 Graph:Upper bound (by flagmatic on 6 vertices): ≈ / ≈ . Graph:Upper bound (by flagmatic on 6 vertices): ≈ c ≈ . c = max x ∈ [0 , ] x (1 − x ) − x = 9 (cid:16) √ − (cid:17) + 6 r (cid:16) √ − (cid:17) . Construction: x − xx Construction G is the following: split vertices into three parts, A of size x , B of size x , and C ofsize 1 − x . Put all possible edges from A to B and from B to C , put also all possible edges inside C to make it an arbitrary tournament. Iterate process inside A and inside B . Graph:Upper bound (by flagmatic on 6 vertices): ≈ c ≈ . c = max x ∈ [0 , ] − x ) x − (1 − x ) = 32 − p √ − q √ − ! . Construction: 10 − xx x Construction G is the following: split vertices into three parts— A of size x , B of size x , and C of size 1 − x . Put all possible edges from C to A , from C to B and all possible edges in parts A and B to make them arbitrary tournaments. Iterate this process inside part C . Graph:Upper bound (by flagmatic on 6 vertices): ≈ c & . c = max 24 y (1 − x − y )1 − x ( xyI + y (1 − x − y ) I + (1 − x − y ) I ) ,I = Z p ( a )(1 − p ( a )) d a, I = Z Z a p ( a ) p ( b )(1 − p ( b )) d b d a,I = Z Z a Z b (1 − p ( c )) p ( b )(1 − p ( a )) d c d b d a and the maximum is taken over all x, y ∈ [0 ,
1] and functions p : [0 , −→ [0 , x, y ∈ [0 ,
1] and p being a polynomial of degree atmost 7.Construction: p ( v )1 − x − yv yx Construction G is the following: split vertices into three parts— A of size x , B of size y , and C of size 1 − x − y . Put all possible edges from A to B , from C to A , and inside part C to make ita transitive tournament. Each vertex v ∈ C can be associated with a number v ∈ [0 ,
1] preservingthe transitive order (we can treat C as a finite subset of [0 ,
1] and so v ∈ [0 ,
1] as well). For eachvertex v ∈ C and w ∈ B , put an edge from v to w independently at random with probability p ( v ).Finally, iterate this process inside part A . Graph:Upper bound (by flagmatic on 5 vertices): (28 + 6 √ / √ / ≈ . + √ Circular graph with parameter (9 + √ / Graph:Upper bound (by flagmatic on 8 vertices): ≈ H has the property that any two vertices v, w ∈ V ( H ) are not adjacent if andonly if N + ( v ) = N + ( w ) and N − ( v ) = N − ( w ), then we may assume that the graphs maximizingthe number of induced copies of H also have this property. In particular, we can consider theinducibility of H in a family of graphs which have no induced copy of T ⊔ A (an edge plus anisolated vertex) or −→ P , simplifying this way the computations in flagmatic.Lower bound: c ≈ . c = max ≤ y ≤ − q ≤ (cid:18) − y − q (cid:19) y (1 − q ) + 24 qy (1 − q ) (cid:18)
11 + q + q − y − q (cid:19) . Construction: a a a a a · · · · · · a + 2 a + 2 a + · · · = 1 Construction consists of a sequence of blobs A i for i ∈ Z , where each A i is an anticlique, and eachpair of vertices from different blobs is joined by an arc pointing to the blob of the larger index.Let blob A i be of the size a | i | , where a + 2 a + 2 a + . . . = 1. Letting the blob sizes decreasegeometrically, i.e., a i = yq i − for i = 1 , , . . . and a = 1 − y − q , we obtain the desired density. Graph:Upper bound (by flagmatic on 6 vertices): ≈ c ≈ . c = max x ∈ [0 ,
1] 85 x (1 − x ) + (1 − x ) − x = 4105 (cid:18) / q √ − − / q
63 + √ (cid:19) . Construction: x − x −→ C ⊙ Construction G is the following: split vertices into two parts— A of size x and B of size 1 − x .Put edges in B to make it an iterated blow-up of −→ C and all possible edges from A to B . Iteratethe process inside A . 12 .24 Graph 24 Graph:Upper bound: 3 / Proof.
Let G be a graph on n vertices maximizing the number of induced copies of H = andintroduce the following equivalence relation in V ( G ): v ∼ w iff N + ( v ) = N + ( w ) and N − ( v ) = N − ( w ). By Lemma 2.1 from [15], since H has the property that v ∼ w iff v and w are not joinedby an edge, we may assume that G satisfies the same property.Denote by T the set of all vertices which belong to equivalence classes of size one, and let t = | T | . Let m be the number of the remaining equivalence classes; denote them by B , . . . , B m ,where b i = | B i | and b ≤ b ≤ . . . ≤ b m . Note that the number of induced copies of H in G doesnot depend on the orientation of edges between vertices in T . Also, by orienting the edges in sucha way that T → B i for every i ∈ [ m ], we won’t decrease the number of induced copies of H . Thefollowing claim gives the orientation of edges between the remaining pairs of equivalence classes. Claim 1.
We can orient the edges between B i and B j for every i < j so that B i → B j withoutdecreasing the number of induced copies of H . Proof.
Consider a partition I ∪ J = [ m −
1] of indices such that B i → B m for i ∈ I and B m → B j for j ∈ J . Let us modify graph G by orienting all edges incident to B j and B m towards B m foreach j ∈ J . Each copy of H removed in this way has exactly one vertex in B m , exactly two verticesin B j for some j ∈ J , and one more vertex which belongs neither to B j nor B m and has outgoingedges to B j . The number of lost copies is then equal to b m X j ∈ J (cid:18) b j (cid:19) t + X i ∈ I,B i → B j b i + X j ′ ∈ J,B j ′ → B j b j ′ . (1)The created copies of H contain exactly two vertices in B m and at least one vertex in some B j for j ∈ J , so the number of created copies is equal to (cid:18) b m (cid:19) t X j ∈ J b j + X i ∈ I,j ∈ J b i b j + X j 2, and G is an extremal graph with b m being maximal and—for this valueof b m —also t being maximal. Claim 2. If m ≥ 2, then t > b m . Proof. Since G is an extremal graph, introducing a single edge in B shall not increase the numberof copies of H . Such an edge removes (cid:0) t (cid:1) copies of H and creates P mi =2 (cid:0) b i (cid:1) copies of H . Therefore,we have (cid:18) t (cid:19) ≥ X i ≥ (cid:18) b i (cid:19) . In particular, t ≥ b m . Moreover, if m > 2, then t > b m , since (cid:0) b (cid:1) ≥ 1. Also, if m = 2 and t = b m , then we can move those two vertices which we just joined by an edge to T ; this way, wewill not decrease the number of copies of H in G , but we will increase the size of T , contradictingour assumptions about the maximality of the graph G . It follows that we must have t > b m .We are ready to give an argument contradicting the maximality of G . For this, remove onevertex from T and add one vertex to B m . This way, we destroy the following number of copiesof H : ( t − m X i =1 (cid:18) b i (cid:19) + m − X i =1 m X j = i +1 b i (cid:18) b j (cid:19) , which can be written in the following way:( t − (cid:18) b m (cid:19) + ( t − m − X i =1 (cid:18) b i (cid:19) + m − X i =1 m − X j = i +1 b i (cid:18) b j (cid:19) + m − X i =1 b i (cid:18) b m (cid:19) . (3)On the other hand, we create the following number of copies of H : b m (cid:18) t − (cid:19) + ( t − m − X i =1 b i + m − X i =1 m − X j = i +1 b i b j , which can be written in the following way: b m (cid:18) t − (cid:19) + ( t − m − X i =1 b i b m m − X i =1 m − X j = i +1 b i b j b m + m − X i =1 ( t − b i b m . (4)Since t − ≥ b m from Claim 2, it is straightforward to see that each summand of (3) is not greaterthan an appropriate summand of (4), and that there is at least one strictly smaller summand (e.g.the last one).It follows that graph G must be of the form T → B . The number of copies of H in G is thenequal to (cid:0) t (cid:1)(cid:0) n − t (cid:1) , which is maximized for t ∈ {⌊ n ⌋ , ⌈ n ⌉} . This gives the desired upper bound forthe inducibility of H .Lower bound: 3/8Construction:Full join from an arbitrary tournament to an anticlique of the same size.14 .25 Graph 25 Graph:Upper bound: 4 / Proof. Note that in each copy of there are exactly two vertices for which outdegree, indegree,and nondegree are equal to 1. Basing on this observation and using AM-GM inequality, we get thefollowing bound on the number of copies of in any n -vertex graph G : N ( , G ) ≤ X v ∈ V ( G ) d + ( v ) d − ( v ) d ′ ( v ) ≤ n ( n − . In particular, I ( ) ≤ lim n →∞ n ( n − / (cid:18) n (cid:19) = 49 . Lower bound: 4/9Construction:Balanced blow-up of −→ C . Graph:Upper bound (by flagmatic on 6 vertices): ≈ c & . c = max 24 y (1 − x − y )1 − x ( xI + (1 − x − y ) I ) ,I = Z p ( a )(1 − p ( a )) d a, I = Z Z a (1 − p ( b )) p ( a )(1 − p ( a )) d b d a and the maximum is taken over all x, y ∈ [0 , 1] and functions p : [0 , −→ [0 , x, y ∈ [0 , 1] and p being a polynomial of degree at most 7.Construction: p ( v )1 − p ( v )1 − x − yv yx G is the following: split vertices into three parts— A of size x , B of size y , and C of size 1 − x − y . Put all possible edges from B to A , from A to C , and inside part C to make ita transitive tournament. Each vertex v ∈ C can be associated with a number v ∈ [0 , 1] preservingthe transitive order (we can treat C as a finite subset of [0 , 1] and so v ∈ [0 , 1] as well). For eachvertex v ∈ C and w ∈ B , put an edge between v and w oriented independently at random, withprobability p ( v ) from v to w and with probability 1 − p ( v ) from w to v . Finally, iterate this processinside part A . Graph:Upper bound (by flagmatic on 6 vertices): ≈ / ≈ . Circular graph with parameter 4 / Graph:Upper bound: c proved in [6].Lower bound: c ≈ . c = max x ∈ [0 , x / x (1 − x )1 − (1 − x ) = 8 − / + 3 / . Construction: x R − x Construction G is the following: split vertices into two parts— A of size x and B of size 1 − x .Let A be a random tournament and put all possible edges from A to B . Finally, iterate the processinside B . Graph:Upper bound: 1 / Proof. Let G be an n -vertex graph with the maximum number of copies of . As adding an arcbetween two non-neighbors may only increase the number of copies of , we may assume that G is a tournament, i.e., d + ( v ) + d − ( v ) = n − v ∈ V ( G ). Note that N ( , G ) ≤ X v ∈ V ( G ) N ( , G − v ) , 16s in each copy of there are two ways to select vertex v in such a way that remaining threevertices form a copy of . Furthermore, for every tournament H on k vertices, it holds2 N ( , H ) + (cid:18) k (cid:19) = 3 N ( , H ) + N ( , H ) = X v ∈ V ( H ) d + ( v ) d − ( v ) ≤ k ( k − N ( , H ) ≤ ( k − k ( k + 1)24 . Applying this inequality for k = n − H = G − v for every v ∈ V ( G ), and plugging to theprevious estimation, we finally get N ( , G ) ≤ n ( n − n − , hence I ( ) ≤ lim n →∞ n ( n − n − / (cid:18) n (cid:19) = 12 . An independent proof can be found in [6].Lower bound: 1/2Construction: Circular graph with parameter 1 / Graph:Upper bound: 1Lower bound: 1Construction: T Transitive tournament. Acknowledgment We would like to thank Jakub Sliacan for his help with the flagmatic software. References [1] J. Balogh, P. Hu, B. Lidický, and F. Pfender: Maximum density of induced 5-cycle is achievedby an iterated blow-up of 5-cycle , European J. Combin. (2016), 47–58.172] J. Balogh, P. Hu, B. Lidický, O. Pikhurko, B. Udvari, and J. Volec: Minimum number ofmonotone subsequences of length 4 in permutations , Combin. Probab. Comput. (2015),658–679.[3] B. Bollobás, Y. Egawa, A. Harris, and G. P. Jin: The maximal number of induced r-partitesubgraphs , Graphs Combin. (1995), 1–19.[4] B. Bollobás, C. Nara, and S.-i. Tachibana: The maximal number of induced complete bipartitegraphs , Discrete Math. (1986), 271–275.[5] J. Brown and A. Sidorenko: The inducibility of complete bipartite graphs , J. Graph Theory (1994), 629–645.[6] D. Burke, B. Lidický, F. Pfender, and M. Phillips: Inducibility of 4-Vertex Tournaments ,manuscript.[7] I. Choi, B. Lidický, and F. Pfender: Inducibility of directed paths , arXiv:1811.03747[8] C. Even-Zohar and N. Linial: A note on the inducibility of -vertex graphs , Graphs Combin. (2015), 1367–1380.[9] J. Fox, H. Huang, and C. Lee: A solution to the inducibility problem for almost all graphs ,manuscript.[10] H. Hatami, J. Hirst, and S. Norine: The inducibility of blowup graphs , J. Combin. Theory Ser.B (2014), 196–212.[11] D. Hefetz and M. Tyomkin: On the inducibility of cycles , J. Combin. Theory Ser. B (2018), 243–258.[12] J. Hirst: The inducibility of graphs on four vertices , J. Graph Theory (2014), 231–243.[13] P. Hu, B. Lidický, F. Pfender, and J. Volec: Inducibility of orientations of C , manuscript.[14] P. Hu, J. Ma, S. Norin, and H. Wu: Inducibility of oriented stars , arXiv:2008.05430[15] H. Huang: On the maximum induced density of directed stars and related problems , SIAM J.Discrete Math. (2014), 92–98.[16] D. Kráľ, S. Norin, and J. Volec: A bound on the inducibility of cycles , J. Combin. TheorySer. A (2019), 359–363.[17] N. Pippenger and M. C. Golumbic: The inducibility of graphs , J. Combin. Theory Ser. B (1975), 189–203.[18] A. Razborov: Flag Algebras , J. Symb. Log. (2007), 1239–1282.[19] E.R. Vaughan: Flagmatic: A tool for researchers in extremal graph theory , further developedby J. Sliacan, available at https://github.com/jsliacan/flagmatic-2.0 .[20] R. Yuster: On the exact maximum induced density of almost all graphs and their inducibility ,J. Combin. Theory Ser. B136