Definability in the embeddability ordering of finite directed graphs
Ad´am Kunos´
Received: date / Accepted: date
Abstract A directed graph G∈D is said to be embeddable into G0∈D if there exists an injective graph homomorphismϕ:G→G0. We consider the embeddability ordering(D,≤) of finite directed graphs, and prove that for every G∈D the set {G,GT}is definable by first-order formulas in the partially ordered set(D,≤), where GTdenotes the transpose ofG. We also prove that the automorphism group of(D,≤) is isomorphic toZ2.
Keywords First-order definability·Directed graph·Embeddability ordering
1 Introduction
In 2009–2010 J. Jeˇzek and R. McKenzie published a series of papers [1–4] in which they have examined (among other things) the first-order definability in the sub- structure orderings of finite mathematical structures with a given type and determined the automorphism group of these orderings. They considered finite semilattices [1], ordered sets [2], distributive lattices [3] and lattices [4].
In this paper we consider (the isomorphism types) of finite directed graphs. Let us consider a nonempty setVand a binary relationE⊆V2. We call the pairG= (V,E)a directed graphor justdigraph. The elements ofV(=V(G))andE(=E(G))are called theverticesandedgesofG, respectively. The directed graphGT:= (V,E−1)is called thetransposeofG, whereE−1denotes the inverse relation ofE. A directed graph is finiteif the number of its vertices is finite. In the papers [1–4] the authors have inves- tigated substructure orderings, meaning thatH≤H0if and only ifHis isomorphic to a substructure ofH0. Differently, we investigate the embeddability ordering, namely
This research was realized in the frames of T ´AMOP 4.2.4. A/2-11-1-2012-0001 “National Excellence Program—Elaborating and operating an inland student and researcher personal support system” The project was subsidized by the European Union and co-financed by the European Social Fund.
Ad´am Kunos´
University of Szeged, Bolyai Institute, Szeged, Aradi v´ertan´uk tere 1, HUNGARY 6720 E-mail: kunosadam@gmail.com
we say that the directed graphGis embeddable intoG0 if there exists an injective mapϕ:V(G)→V(G0)such that(v1,v2)∈E(G)implies(ϕ(v1),ϕ(v2))∈E(G0). It is obvious that isomorphic digraphs are indistinguishable in terms of embeddability.
So from now on by a given digraphGwe always mean its isomorphism type, for we intend to work with embeddability. LetD denote the set of isomorphism types of finite digraphs. ForG,G0∈D, byG≤G0we mean thatGis embeddable intoG0. It is easy to verify that(D,≤)is a partially ordered set. It is also easy to see that the map G7→GT (G∈D)is a non-trivial automorphism of the poset(D,≤). We will prove that there is no other non-trivial automorphism of(D,≤).
Let (A,≤) be an arbitrary poset. An n-ary relationR is said to be definable in(A,≤)if there exists a first-order formulaΨ(x1,x2, . . . ,xn) with free variables x1,x2, . . . ,xnin the language of partially ordered sets such that for anya1,a2, . . . ,an∈ A,Ψ(a1,a2, . . . ,an)holds in(A,≤)if and only if(a1,a2, . . . ,an)∈R. A subset of A is definable if it is definable as a unary relation. An elementa∈A is said to be definable if the set{a}is definable.
In the poset(D,≤)letG≺G0 denote thatG0coversG. Obviously≺is a defin- able relation in(D,≤). It is easy to see that if there exists an automorphism of an arbitrary poset(A,≤)that maps the elementa∈A tob∈A thenaandbare in- distinguishable in(A,≤)with first-order formulas. This tells us, considering the fact thatG→GT is a non-trivial automorphism of(D,≤), that the “best” we can prove is that the set ˜G:={G,GT}is definable for everyG∈D. We prove this in the next section.
2 The definability of the sets{G,GT}
Definition 2.1 Let us introduce the following digraphs:
E1:V(E1) ={v1},E(E1) =/0, L1:V(L1) ={v1},E(L1) ={(v1,v1)}, E2:V(E2) ={v1,v2},E(E2) =/0, and I2:V(I2) ={v1,v2},E(I2) ={(v1,v2)}.
Lemma 2.2 The digraphs E1, L1, E2, I2are definable.
Proof E1 is the unique digraphX ∈D for whichX ≤G holds for everyG∈D. There are two elements covering E1, namely L1(=LT1) andE2(=E2T), hence the set{L1,E2} is definable. In this set only L1 has a unique cover so L1 andE2 are distinguishable, which implies that ˜L1={L1}is definable and so is ˜E2={E2}.I2is the unique element among the covers ofE2which has 4 covers and does not coverL1
(see Fig. 1). ut
Definition 2.3 For a positive integern, we say thatG∈Dis on levelnif|V(G)|+
|E(G)|=n. LetSndenote the set of all digraphs on leveln.
It is an easy observation that ifG≺G0thenG0is exactly one level aboveG, so our definition corresponds to the levels of the Hasse diagram of(D,≤)(see Fig. 1).
Lemma 2.4 The setSnis definable for every positive integer n.
Definability in the embeddability ordering of finite directed graphs 3
E1
O2 E4
I2 E3
L2
E2 L1
F G H
E I J
T2
U2
V2
W2
Z2 A3
B3 C3D3 E3 F3 G3H3 I3 J3 K3L3 M3
N4 H5 O4 PI54 JQ54 R4 K5 S4 N6
O6
Fig. 1 The “bottom” part of the Hasse diagram of(D,≤).
Proof S1was already defined in Lemma 2.2.Sn+1can be defined recursively:Sn+1
is the set of digraphsX∈Dfor whichZ≺Xholds for someZ∈Sn. ut Definition 2.5 We say that the digraphXisn level under Zif there existZ1, . . . ,Zn∈ D such thatZZ1 · · · Zn=X. Similarly, we say that the digraphX isn level above Yif there existY1, . . . ,Yn∈Dsuch thatY≺Y1≺ · · · ≺Yn=X.
Definition 2.6 LetEn(n=1,2, . . .)be the “empty” digraph withnvertices:V(En) = {v1,v2, . . . ,vn},E(En) =/0. LetFn(n=1,2, . . .)be the “full” digraph withnvertices:
V(Fn) ={v1,v2, . . . ,vn},E(Fn) =V(Fn)2. LetTn={G∈D:|V(G)|=n}be the set of all digraphs havingnvertices.
Lemma 2.7 The digraphs En, Fnand the setTnare definable for every positive inte- ger n.
Proof The setI ={En:n∈ {1,2, . . .}}is definable, because its elements are those digraphsX∈D for whichL1X andI2X. InI we haveE1≺E2≺E3≺. . ., therefore it is easy to see thatEn(=EnT)is definable for everyn.Tncontains exactly those digraphsX ∈D for whichEn≤X andEn+1X, so it is definable.Fnis the digraphX∈Dwhich hasnvertices andX≺Zimplies thatZhasn+1 vertices. ut Definition 2.8 Let us set notations for the following digraphs (see Fig. 2):
O2:V(O2) ={v1,v2},E(O2) ={(v1,v2),(v2,v1)},
O2 A I3 L2 L2+
Fig. 2 The digraphsO2,A,I3,L2,L2+.
A:V(A) ={v1,v2,v3},E(A) ={(v1,v3),(v2,v3)}, I3:V(I3) ={v1,v2,v3},E(I3) ={(v1,v2),(v2,v3)}, L2:V(L2) ={v1,v2},E(L2) ={(v1,v1),(v2,v2)}, and L2+:V(L2+) ={v1,v2},E(L2+) ={(v1,v1),(v2,v2),(v1,v2)}.
Lemma 2.9 The digraphs O2, I3, L2, L2+and the setA are definable.˜ Proof O2(=OT2)is the maximal digraphX that has 2 vertices andL1X.
LetG∈D be the following digraph:V(G) ={v1,v2,v3},E(G) ={(v1,v2)}. Then G(=GT)is definable since it is the only element covering bothI2andE3.
Now the set ˜A∪I˜3turns out to be definable, for it contains exactly those digraphs X ∈D for whichG≺X,O2X,E4X andL1X hold. Now ˜I3={I3}is the unique digraphX ∈A˜∪I˜3 for which there exists X ≺Z such thatW ≺Z implies X=W (with the notation to be introduced in Definition 2.10,Z=O3), meaningZ covers onlyX. From this we also get that ˜Ais definable because ˜A= (A˜∪I˜3)\I˜3. The digraph L2 is the maximal X ∈D that has 2 vertices and for which I2X.
Finally, the digraphL2+ is the onlyX ∈D having 2 vertices, being on level 5 for
whichL2≤X. ut
Definition 2.10 LetIn,On,Ln(n=2,3, . . .)denote the following digraphs:
V(In) =V(On) =V(Ln) ={v1,v2, . . . ,vn}, E(In) ={(v1,v2),(v2,v3), . . . ,(vn−1,vn)},
E(On) =E(In)∪ {(vn,v1)}={(v1,v2),(v2,v3), . . . ,(vn−1,vn),(vn,v1)},and E(Ln) ={(v1,v1),(v2,v2), . . . ,(vn,vn)}.
Lemma 2.11 The digraphs Ln, On, In(n=2,3, . . .)are definable.
Proof Ln(=LTn)is the unique digraphXon level 2nthat hasnvertices and for which I2X.
We define the digraphsOn(=OTn)andIn(=InT)together, recursively.O2andI2were already defined in Lemma 2.9 and Lemma 2.2, respectively. Suppose thatO2,I2, . . . , On,Inhave already been defined. ThenOn+1is the unique digraphXthat:
– hasn+1 vertices,
– In≤X,L1X,O2X,OnX,
I5 O6 L6
Fig. 3 The digraphsI5,O6andI6.
– is on level 2n+2, and
– there exists noZ∈A˜for whichZ≤X.
Finally, the digraphIn+1is the only elementXfor whichX≺On+1. ut Definition 2.12 LetGbe an arbitrary finite directed graph with no loops. LetL(G) denote the digraph that we get from Gby adding loops to every vertex. For a set G ⊆Dof finite digraphs with no loops letL(G):={L(G):G∈G}.
Definition 2.13 For an arbitraryG∈DletM(G)denote the digraph we get by leav- ing all the loops out fromG. For a setG⊆Dof finite digraphs putM(G):={M(G): G∈G}.
Lemma 2.14 LetG ⊆Dbe a definable set of finite digraphs with no loops. Then the setL(G)is definable.
Proof We define the binary relation
α={(G,M(G)):G∈D} (1)
as the set of pairs(X,Y)∈D2for whichY is the maximal digraph withY ≤X and L1Y. NowL(G)is the set of those digraphsXfor which there existsY ∈G such
thatXis the maximal digraph with(X,Y)∈α. ut
Lemma 2.15 LetG ⊆Dbe a definable set of finite digraphs. Then the setM(G)is definable.
Proof The setM(G)is definable as the set of those digraphsXfor which there exists Y ∈G such thatXis the maximal digraph withX≤YandL1X. ut Definition 2.16 LetOn,Lbe the following digraph (see Fig. 4):V(On,L) ={v1,v2, . . . ,vn}, E(On,L) =E(On)∪ {(v1,v1)}, which means that
E(On,L) ={(v1,v1),(v1,v2),(v2,v3), . . . ,(vn−1,vn),(vn,v1)}.
Lemma 2.17 The digraphs On,L(n=2,3, . . .)are definable.
Fig. 4 The digraphO3,L.
Fig. 5 The digraphOL(3,4,5).
Proof On,L(=OTn,L)is the unique digraphX on level 2n+1 for whichOn≤X and
L1≤X. ut
Definition 2.18 For arbitrary finite directed graphsG1,G2, . . . ,Gnlet us denote their disjoint uniun byG1∪˙ G2∪˙. . .∪˙ Gn=S˙n
i=1Gi, as usual.
Definition 2.19 Letn1,n2, . . . ,nkbe pairwise distinct integers greater than 1. Let O(n1,n2, . . . ,nk) =[˙k
i=1Oni; OL(n1,n2, . . . ,nk) =[˙k
i=1Oni,L. Lemma 2.20 The digraphs O(n1,n2, . . . ,nk)and OL(n1,n2, . . . ,nk)are definable.
Proof O(n1,n2, . . . ,nk)(= (O(n1,n2, . . . ,nk))T)is the unique digraphX havingn1+ n2+· · ·+nkvertices, being on level 2(n1+n2+· · ·+nk)for whichOni ≤X (i= 1,2, . . . ,k)and there exists noZ∈A˜such thatZ≤X.
OL(n1,n2, . . . ,nk)(= (OL(n1,n2, . . . ,nk))T)is the unique digraph X being on level 2(n1+n2+· · ·+nk) +kfor whichO(n1,n2, . . . ,nk)≤X,Lk≤X andOni,L≤X(i=
1,2, . . . ,k). ut
Definition 2.21 Letiand jbe different positive integers both bigger than 2. Let us define the digraphOi,j,L1(see Fig. 6) the following way. LetV(Oi) ={v1, . . . ,vi}and V(Oj) ={v01, . . . ,v0j}. Now letV(Oi,j,L1) =V(Oi∪O˙ j),
E(Oi,j,L1) =E(Oi∪˙ Oj)∪ {(v1,v1),(v01,v01),(v1,v01)}.
LetOi,j,L2:V(Oi,j,L2) =V(Oi,j,L1),E(Oi,j,L2) =E(Oi,j,L1)∪ {(v01,v1)}and finally let Oi,j,1=M(Oi,j,L1),Oi,j,2=M(Oi,j,L2).
In words, Oi,j,L1 consists of two circles which are connected by an L2+ (see Fig. 6).
Lemma 2.22 Let i and j be different positive integers both bigger than2. Then the setsO˜i,j,L1,O˜i,j,L2,O˜i,j,1andO˜i,j,2are definable.
O3,4,L1 O3,4,L2
Fig. 6 O3,4,L1,O3,4,L2
X W
Fig. 7 A digraphX∈O4,5,1+ and a digraphW∈O4,5,2+ .
Proof O˜i,j,L1is the set of those digraphsXthat:
– havei+jvertices,
– are on level 2(i+j) +3, and – OL(i,j)≤X,L2+≤X,L3X. O˜i,j,L2consists of those digraphsXthat:
– are on level 2(i+j) +4, – F2≤X, and
– there existsZ∈O˜i,j,L1such thatZ≤X.
Finally, the definability of the digraphs ˜Oi,j,1, ˜Oi,j,2follows from Lemma 2.15. ut Definition 2.23 LetOi,j,1+ denote the set of those digraphs coveringZ inD which are obtained by adding an edge toZthat is not a loop, whereZ∈O˜i,j,1. Similarly, let Oi,+j,2denote the set of those digraphs coveringZinDwhich are obtained by adding an edge toZthat is not a loop, whereZ∈O˜i,j,2. (see Fig. 7).
Lemma 2.24 The setsOi,+j,1,Oi,+j,2are definable for every distinct positive integers i,j both bigger than two.
Proof Oi,+j,1consists of those digraphsXfor whichZ≺Xfor someZ∈O˜i,j,1,L1X andEi+j+1X. Similarly,Oi,j,2+ is the set of those digraphsX for whichZ≺X for
someZ∈O˜i,j,2,L1XandEi+j+1X. ut
Definition 2.25 For a finite directed graphG∈D, letA(G)be the set of those di- graphs that can be obtained fromGby reversing some edges.
Definition 2.26 Let u,v,w be 3 distinct vertices of a directed graphG∈D. Let G|{u,v,w}denote the digraph induced by the verticesu,v,w. Let[u,v,w]Gdenote the digraph for whichV([u,v,w]G):=V(G|{u,v,w})and
E([u,v,w]G):=E(G|{u,v,w})∩ {(u,v),(v,u),(v,w),(w,v)}.
Lemma 2.27 Let G∈Dbe a weakly connected digraph with vertices v1, . . . ,vnfor which O2G. Then
{G,GT}={X∈A(G):([vi,vj,vk]X=I3)⇔([vi,vj,vk]G=I3),
([vi,vj,vk]X∈A)˜ ⇔([vi,vj,vk]G∈A)}.˜ (2) Proof We can suppose thatGhas an edge because otherwiseGmust have a single vertex which is a trivial case. We can also suppose that(v1,v2)∈E(G), for it is only a matter of notation. LetH denote the right-hand side of (2). It is clear that for any X∈H, there is exactly one edge between the verticesv1andv2sinceH⊆A(G)and O2G. Let us suppose first that for anX∈H, similarly toG,(v1,v2)∈E(X)holds.
We claim thatX =G. Let us consider an arbitrary pair of verticesv0,v00∈V(X)that are connected by an edge inX. SinceGis weakly connected there exists a series
v1=w1,v2=w2,w3, . . . ,wk−2,v0=wk−1,v00=wk
of pairwise distinct vertices such that for all j∈ {1,2, . . . ,k−1}there is exactly one edge between the edgeswj,wj+1. Let us consider the neighbouring vertices in this series. We know that(w1,w2)∈E(G),E(X). Observe that the direction of the edge between the verticesw2,w3inXis determined by the conditions
([w1,w2,w3]X=I2)⇔([w1,w2,w3]G=I2), ([w1,w2,w3]X∈A)˜ ⇔([w1,w2,w3]G∈A),˜
moreover it has the same direction inX as inGand so on, the direction of the edge between the verticeswj,wj+1is determined by the conditions
([wj−1,wj,wj+1]X=I2)⇔([wj−1,wj,wj+1]G=I2), ([wj−1,wj,wj+1]X∈A)˜ ⇔([wj−1,wj,wj+1]G∈A)˜
and it has the same direction inX andG. We proved that the direction of the edge connecting verticesv0,v00is the same inXas inG, therefore we provedX=G. If we suppose the converse:(v2,v1)∈E(X)thenXT =Gby the previous case. ut Definition 2.28 For pairwise distinct positive integersi, j,kgreater than 2 we de- fine the digraphOi→j→k(see Fig. 8) the following way. LetV(Oi) ={v1,v2, . . . ,vi}, V(Oj) ={v01,v02, . . . ,v0j}andV(Ok) ={v001,v002, . . . ,v00k}. Now letV(Oi→j→k) =V(Oi∪˙ Oj∪˙ Ok)and
E(Oi→j→k) =E(Oi∪˙ Oj∪˙ Ok)∪ {(v1,v1),(v01,v01),(v001,v001),(v1,v01),(v01,v001)}.
Oi→j←kis defined similarly, by modifying the definition ofOi→j→knaturally by re- placing(v01,v001)with(v001,v01).
O3→4→5 O3→4←5
Fig. 8 The digraphsO3→4→5andO3→4←5.
In words,Oi→j→k andOi→j←k consist of three disjoint circlesOi,Oj,Ok with one loop on each that are connected in the wayi→ j→kandi→ j←kaccording to the sizes of the circles they are on (see Fig. 8).
Lemma 2.29 Let i, j, k be pairwise distinct positive integers bigger than2. Then the setsO˜i→j→kandO˜i→j←kare definable.
Proof O˜i→j→kis the set of those digraphsXthat:
– havei+j+kvertices, – OL(i,j,k)≤X,
– are on level 2(i+j+k) +5,
– there existZ∈O˜i,j,L1andW ∈O˜j,k,L1such thatZ,W≤X, and – L(I3)≤X.
We can define ˜Oi→j←k almost the same way as ˜Oi→j→k, the difference is that we replace the conditionL(I3)≤Xby: there existsZ∈L(A)˜ for whichZ≤X. ut Definition 2.30 Let Gbe a finite directed graph withn vertices and no loops and letv= (v1,v2, . . . ,vn)be a vector containing all the vertices ofGin some fixed or- der. Let us define the digraphK(G,v)the following way. Let us consider the circles On+1,On+2, . . . ,O2nwithV(On+i) ={vi,j: 1≤j≤n+i}. Now let
V(K(G,v)) =V( [˙
1≤i≤n
On+i),
E(K(G,v)) =E( [˙
1≤i≤n
On+i)∪ {(vi,1,vj,1):(vi,vj)∈E(G)}
∪ {(vi,1,vi,1): 1≤i≤n}.
In words we getK(G,v)fromGby adding loops and big, differently sized circles to all the vertices ofG(see Fig. 9).
Example 1 Figure 9 showsK(I3,v)as an example (withv= (v1,v2,v3)).
I3 K(I3,v)
v1 v2 v3
Fig. 9 I3and a correspondingK(I3,v).
This type of graph will be useful because in this graph—thanks to the big circles—
we can distinguish between the vertices ofGwhich will allow us to define which pairs of vertices are connected with how many edges, so we will be able to define the set A(G).
Lemma 2.31 For an arbitrary weakly connected finite digraph G∈Dwith no loops the setG˜={G,GT}is definable.
Proof First we consider all the pairs of vertices that have edges in both directions between them and we clear one of the two edges for each such pair. We may get different digraphs clearing different edges but as for the proof it does not matter which one we take so let us fix one such digraphG0withO2G0for the rest of the proof.
Letv= (v1,v2, . . . ,vn)be a fixed vector of the vertices ofG. We first define the set H1:={K(G0,v),K((G0)T,v)}.
This set contains exactly those digraphsX that:
(1) have(n+1) + (n+2) +· · ·+ (n+n)
=n(3n+1)2
vertices, (2) OL(n+1,n+2, . . . ,n+n)≤X,
(3) Ln+1X,
(4) if there is no edge between the verticesvi,vj inG0, then for allZ∈O˜n+i,n+j,1, ZXholds,
(5) if there is an edge between the verticesvi,vjinG0then there is anZ∈O˜n+i,n+j,L1
for whichZ≤Xbut for allW∈On+i,n+j,1+ ,WX holds,
(6) if[vi,vj,vk]G0=I3, there exists aZ∈O˜(n+i)→(n+j)→(n+k)for whichZ≤X, and (7) if[vi,vj,vk]G0∈A˜there exists aZ∈O˜(n+i)→(n+j)←(n+k)for whichZ≤X.
The conditions (1)–(3) ensure the structure of “big” circles, determine the number of loops. The conditions (4)–(5) tell how many and what kind of edges are to be drawn between the “big circles”. The conditions (1)–(5) define the set
{K(Z,v):Z∈A(G0)}
while the conditions (6)–(7), by Lemma 2.27, choose the setH1from it.
It is easy to see that the conditions (1)–(5) can similarly define the set H2:={K(Z,v):Z∈A(G)}
G H∈O(G) Fig. 10 A digraphGand a correspondingH∈O(G).
with the following condition added (and writingGinstead ofG0in every other con- dition, naturally):
(8) if there are edges between the verticesvi,vj in both directions then there exists a digraphZ∈O˜n+i,n+j,L2for whichZ≤X but for anyW ∈On+i,n+j,2+ ,W X holds.
Now the set
H3:={K(G,v),K((G)T,v)}
can be defined the following way. It consists of those digraphsX that:
(9) X∈H2, and
(10) there exists a digraphZ∈H1for whichZ≤X.
Finally the setH4={L(G),L(GT)}consists of thoseXthat:
(11) havenvertices, (12) Ln≤X,
(13) X≤Zfor someZ∈H3, and
(14) X is maximal with the previous properties.
Finally we can prove the definability of ˜Gusing Lemma 2.15, forM(H4) ={G,GT}.
u t LetG1,G2, . . . ,Gnbe the weakly connected components of an arbitraryG∈D. Letv1∈G1,v2∈G2, . . . ,vn∈Gnbe arbitrary but fixed vertices. LetN=|V(G)|.
Let us build the digraphH the following way. We add the vertices v01, v02, . . . ,v0N to the digraphGso that they constitute anON circle. We make this digraph, having n+1 weakly connected components, weakly connected by adding the edges(v01,v1), (v02,v2), . . . ,(v0n,vn). It is clear that this contruction depends only on the choice of the vi’s.
Definition 2.32 Let us denote byO(G)the set of all digraphsHthat can be created this way.
Example 2 In Figure 10 we can see a digraphGhaving 2 weakly connected compo- nents and anH∈O(G).
Lemma 2.33 For every finite directed graph G∈Dwith no loops the set{G,GT}is definable.
♂6 ♂L6
Fig. 11 ♂6and♂L6.
Proof We only need to deal with thoseG∈Dthat have more than one weakly inde- pendent components, for we have dealt with the other case before. Let us consider a digraphH∈O(G). SinceHis weakly connected and does not have loops, we know that ˜H is definable by Lemma 2.31. LetH0 be the digraph that we get fromH by adding loops to all the vertices corresponding toG(that do not belong to the “big circle”).
Let|V(G)|=N. ˜H0is the set of those digraphsX∈Dsuch that – there existsZ∈H˜ such thatZisNlevel underX, and – LN≤X,ON,LX.
InH0there are loops exactly on those vertices that correspond toGwhich allows us to defineL(G): it is the set of those digraphs˜ Xsuch that:
– X≤Zfor someZ∈H˜0,
– X hasNvertices andLN≤X, and
– X is maximal with the previous properties.
We are done since ˜G=M(L(G))˜ is definable by Lemma 2.15. ut Definition 2.34 Let ♂n denote the following digraph (see Fig. 11). LetV(On) = {v1, . . . ,vn}and let us define♂nwithV(♂n) =V(On)∪ {v}andE(♂n) =E(On)∪ {(v1,v)}. Let♂Lndenote the digraph that is obtained from♂nby adding a loop the following way:
E(♂Ln) =E(♂n)∪ {(v,v)}.
Since♂nis weakly connected and has no loops, ˜♂nis definable by Lemma 2.31.
Lemma 2.35 The digraphs♂˜Ln(n=2,3, . . .)are definable.
Proof ♂˜Lnis the set of those digraphsXsuch that:
– Z≺Xfor someZ∈♂˜n, and
– L1≤X andOn,LX. ut
Lemma 2.36 For an arbitrary weakly connected finite directed graph G∈Dthe set {G,GT}is definable.
G♂ Fig. 12 G
♂for a givenGwith a given order of vertices.
Proof Letv1,v2, . . . ,vnbe the vertices ofG. Let us consider the circles{On+i}ni=1so thatV(On+i) ={vn+i,j: 1≤j≤n+i}. We define the digraphG♂with
V(G♂) =V
G∪˙ [˙n
i=1On+i
,and
E(G♂) =E
G∪˙ [˙n
i=1On+i
∪ {(vn+i,1,vi): 1≤i≤n}.
In words, we add big circles toGand connect them to the vertices ofGwith edges pointing inG’s direction. (G♂ depends on the order of v1, v2, . . . ,vn. We do not emphasize this in the notation because we need this structure only once and here it is enough to think of any fixed order of the vertices.) An example of the construction is shown in Figure 12.
SinceM(G♂)is weakly connected and has no loops, the setM(G˜♂)is definable by Lemma 2.31. We can suppose thatGhas at least one loop because we dealt with the other case in Lemma 2.31. Letvi1,vi2, . . . ,vik be the list of vertices ofGwith loops. ˜G♂is now the set of those digraphsXsuch that:
– X isklevel above someZ∈M(G˜♂), and
– for every 1≤l≤kthere existsZ∈♂˜Ln+il for whichZ≤X. LetG0
♂=G∪˙
˙ Sn
i=1On+i
. Then ˜G0♂is the set of thoseXthat:
– arenlevel underZfor someZ∈G˜♂,
– Z∈♂˜n+iimpliesZXfor every 1≤i≤n, and – On+i≤X for every 1≤i≤n.
Finally, ˜Gconsists of thoseXthat:
– X≤Zfor someZ∈G˜0♂, – havenvertices andLk≤X, and
– Z≤Xfor someZ∈M(G).˜ ut
Lemma 2.37 For a digraph G∈D that has at least one loop in every weakly con- nected component, the set{G,GT}is definable.
Proof We can still suppose thatGhas more than one weakly independent compo- nents, for we have dealt with the other case previously. Let us consider a digraph H∈O(G). SinceHis weakly connected, the set ˜His definable by Lemma 2.36. Let n(>1)be the number of weakly connected components ofGandmbe the number of its vertices. LetH0be the digraph that we get fromHby leaving the edges out that connect the “big circle” to the components ofG, meaning we leavenedges out and getH0=G∪˙ Om. ˜H0contains exactly those digraphsX such that
– X isnlevel underZfor someZ∈H,˜ – there is noZ∈♂˜mfor whichZ≤X, and – Om≤X.
H0consists ofn+1 weakly connected components from which exactlynhave loops in them, exactly the components corresponding toG. Finally ˜Gis the set of those digraphsXsuch that:
– X≤Zfor someZ∈H˜0,
– X hasmvertices and is on the same level asG, – X has the same number of loops asGhas, and
– Z≤Xfor someZ∈M(G)˜ (M(G)˜ is definable by Lemma 2.33). ut Theorem 2.38 For all G∈D, the set{G,GT}is definable.
Proof We can still supposeGhas more than one weakly independent components for the same reason as above. Let us consider a digraphH∈O(G)again. By Lemma 2.36 the set ˜His definable. We can define ˜H0just as in the proof of Lemma 2.37. LetGL denote the digraph that consists of those weakly connected components ofG that contain a loop. By Lemma 2.37 the set ˜GLis definable. ˜Gis the set of those digraphs Xsuch that:
– X has the same number of vertices and is on the same level asG, – X≤Zfor someZ∈H˜0,
– Z≤Xfor someZ∈M(G), and˜
– Z≤Xfor someZ∈G˜L. ut
3 The automorphism group of(D,≤)
So far we know two automorphisms of(D,≤), namely the trivial one andG7→
GT. In this section we prove that there is no other, meaning that the automorphism group of(D,≤)is isomorphic toZ2.
Lemma 3.1 GT ≤G∪˙ Onimplies G=GT for every finite digraph G and integer 2≤n.
Proof Our first easy observation is thatX≤OnimpliesX=XT. Let us denote the weakly connected components ofGby{Ga}a∈A. LetA=B∪˙ Csuch thatb∈Bif
a∈A
Ga ≤
a∈A
Ga ∪˙ On
˙ [
a∈A
GTa ≤ [˙
a∈A
Ga
!
∪˙ On
˙ [
b∈B
GTb
!
∪˙ [˙
c∈C
GTc
!
≤ [˙
b∈B
Gb
!
∪˙ [˙
c∈C
Gc
!
∪˙ On which obviously implies
˙ [
c∈C
GTc ≤ [˙
c∈C
Gc
!
∪˙ [˙
b∈B
Gb
!
∪˙On
| {z }
=:X
. (3)
If there exists ac∈C for whichGTc ≤X, then GTc ≤On, forX consists of weakly connected components embeddable intoOn. Then, according to our first observation, GTc =Gc which meansGc≤On, a contradiction. This means there is noc∈Cfor whichGTc ≤Xso from (3) we deduce
˙ [
c∈C
GTc ≤[˙
c∈C
Gc, and
˙ [
c∈C
Gc
!T
≤[˙
c∈C
Gc.
By transposing both sides the direction of the embeddability stays the same obviously, but we get the converse, implying
˙ [
c∈C
Gc
!T
=[˙
c∈C
Gc. Using our first observation once more, we obtain
˙ [
b∈B
Gb
!T
=[˙
b∈B
GTb=[˙
b∈B
Gb. Finally, putting together what we have we get
G= [˙
a∈A
Ga= [˙
b∈B
Gb
!
∪˙ [˙
c∈C
Gc
!
= [˙
b∈B
Gb
!T
∪˙ [˙
c∈C
Gc
!T
=GT. u t
Fig. 13 The digraphX4.
Definition 3.2 Let us callXn(see Fig. 13) the digraph with
V(Xn) ={v,v1,v2, . . . ,vn}, E(Xn) ={(v1,v),(v2,v), . . . ,(vn,v)}.
Theorem 3.3 The poset(D,≤)has exactly two automorphisms, namely the trivial and the one that maps every digraph to its transpose. Consequently, the automor- phism group of(D,≤)is isomorphic toZ2.
Proof It is easily seen that an automorphism can only move the elements ofDinside definable sets, therefore from Theorem 2.38 it follows that it either does not move an element or maps it to its transpose. Let us consider an automorphismϕ:D7→Dfor which there existsG∈Dsuch thatG6=GT andϕ(G) =G. We must show thatϕis the identity function. This can be done by showing that addingGto the language of partially ordered sets as a constant results in every element ofDbecoming definable.
So let us addGto the language of partially ordered sets as a constant and pick an arbitraryF∈D that is not isomorphic to its transpose (those digraphs that are iso- morphic to their transposes are definable by Theorem 2.38). Our goal will be to show thatFis definable.
LetV(G) ={v1,v2, . . . ,vn}. Lemma 3.1 lets us defineG∪˙On+1as the unique element from the definable set
{G∪˙ On+1,(G∪˙ On+1)T =GT ∪˙On+1} thatGis embeddable into. Let us use the notation
V(G∪˙ On+1) =V(G)∪ {v01,v02, . . . ,v0n+1}.
We create a digraphG0(see Fig. 14) by adding edges toG∪˙ On+1as follows:
E(G0) =E(G∪˙ On+1)∪ {(v01,v1),(v02,v1),(v03,v1), . . . ,(v0n+1,v1)}. (4) NowG0 is definable as the unique element of the set {G0,(G0)T} into which G∪˙ On+1is embeddable. Xn+1is the unique element from the set{Xn+1,(Xn+1)T}that is embeddable intoG0so it is definable too.Ais the unique element from the set{A,AT} that is embeddable intoXn+1. So far we have proven thatAis definable.
Now we do the same as above, but backwards. Letmbe the number of vertices ofF.
Xm+1is the unique element in the set{Xm+1,(Xm+1)T}thatAis embeddable into. Let F0be created fromFanalogously to howG0was created fromGin (4) (see Fig. 14).
NowF0is the only element from the set{F0,(F0)T}thatXm+1is embeddable into.
Next,F∪˙ Om+1is the only element from the definable set {F∪˙Om+1,(F∪˙ Om+1)T =FT ∪˙ Om+1}
O3 O03
Fig. 14 The digraphO3and a correspondingO03.
that is embeddable intoF0. Finally, by Lemma 3.1,Fis definable as the only element from the set{F,FT}that is embeddable intoF∪˙Om+1. ut
Acknowledgements The author is very thankful to his supervisor, Mikl´os Mar´oti, who introduced him to the problem investigated in this paper and supported him even from overseas.
References
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