On the criteria of D-planarity of a tree
Let $T$ be a tree with a fixed subset of vertices $V^{\ast}$ such that there is a cyclic order $C$ on it and all terminal vertices are contained in this set. Let $D^{2} = \{(x,y) \in \rr^{2} | x^{2} + y^{2} \leq 1 \}$ be a closed oriented 2--dimensio…
Authors: E. Polulyakh, I. Yurchuk
On the riteria of D -planarit y of a tree. P oluly akh E., Y ur h uk I. Institute of Mathematis of Ukrainian national aadem y of sienes, Kyiv Abstrat. Let T b e a tree with a xed subset of v erties V ∗ su h that there is a yli order C on it and all terminal v erties are on tained in this set. Let D 2 = { ( x, y ) ∈ R 2 | x 2 + y 2 ≤ 1 } b e a losed orien ted 2dimensional disk. The tree T is alled D - planar if there exists an em b edding ϕ : T → R 2 whi h satises the follo wing onditions ϕ ( T ) ⊆ D 2 , ϕ ( T ) ∩ ∂ D 2 = ϕ ( V ∗ ) and if ♯V ∗ ≥ 3 then a yli order ϕ ( C ) of ϕ ( V ∗ ) oinides with a yli order whi h is generated b y the orien tation of ∂ D 2 ∼ = S 1 . W e obtain a neessary and suien t ondition for T to b e D -planar. Keyw ords . D planar tree, a yli order, a on v enien t relation. Intr odution. It is kno wn that an y tree is planar (i.e., an b e em b edded in to a plane), more o v er, in [7℄ it w as pro v ed that an y ro oted tree with n v erties an b e em b edded as a plane spanning tree on n p oin ts of a plane, with the ro ot b eing mapp ed on to an arbitrary sp eied p oin t of them. There are similar results for ro oted star forests [8℄. Man y authors w ere in terested in dieren t t yp es of em b edding of trees in to a disk with extra onditions. F or example, in [9, 10℄ author used a sp eial t yp e of em b edding of tree in to a disk for onstruting a Morse funtion on it. In this pap er w e will onsider an em b edding of a tree T in to D 2 (i.e., a losed orien ted 2dimensional disk) su h that a xed subset of its v erties V ∗ whi h on taines all terminal v erties maps to a b oundary of D 2 in a sp eial w a y and T \ V ∗ maps in to I ntD 2 . The existene of su h em b edding is a part of a solution of one top ologial prob- lem: the realization of a nite graph as in v arian t of pseudoharmoni funtions dened on a disk. Namely , a on tin uous funtion f : D 2 → R is alled a pseudo- harmoni funtion if f | ∂ D 2 is a on tin uous funtion with a nite n um b er of lo al extrema and f | In t D 2 has a nite n um b er of ritial p oin ts and ea h of them is a saddle p oin t (in the neigh b orho o d of it f has a represen tation as Rez n + const , z = x + iy and n ≥ 2 ). In [6℄ the top ologial in v arian t of su h funtions is onstruted, in partiular, it onsists of a yle γ with sp eial prop erties and the omplemen t to it is a disjoin t union of trees. The riteria of top ologial equiv alene of su h funtions is form ulated in terms of their in v arian t. In Setion 1 w e will desrib e some prop erties of a D -planar tree. They will b e useful for the pro of of Theorem 2.1 whi h giv es the riteria of D -planarit y of a tree. In this setion w e also study some prop erties of dieren t t yp es of relations on nite sets. 1 2 In Setion 2 the riteria of D -planarit y of a tree will b e pro v ed. Its pro of has a top ologial nature. W e are sinerely grateful to V.V. Shark o, S.I. Maksymenk o and I. Y u. Vlasenk o for useful disussions. 1. The utility resul ts. 1.1. Prop erties of trees em b edded in to t w o-dimensional disk. Let T b e a tree with a set of v erties V and a set of edges E . Supp ose that T is non degenerated ( has at least one edge). Denote b y V ter a set of all v erties of T su h that their degree equals to 1. Let us assume that for a subset V ∗ ⊆ V the follo wing ondition holds true (1) V ter ⊆ V ∗ . Let also ϕ : T → R 2 is an em b edding su h that (2) ϕ ( T ) ⊆ D 2 , ϕ ( T ) ∩ ∂ D 2 = ϕ ( V ∗ ) . Lemma 1.1. A set R 2 \ ( ϕ ( T ) ∪ ∂ D 2 ) has a nite numb er of onne te d omp onents U 0 = R 2 \ D 2 , U 1 , . . . , U m , and for every i ∈ { 1 , . . . , m } a set U i is an op en disk and is b ounde d by a simple lose d urve ∂ U i = L i ∪ ϕ ( P ( v i , v ′ i )) , L i ∩ ϕ ( P ( v i , v ′ i )) = { ϕ ( v i ) , ϕ ( v ′ i ) } wher e L i is an ar of ∂ D 2 suh that the verti es ϕ ( v i ) and ϕ ( v ′ i ) ar e its endp oints, and ϕ ( P ( v i , v ′ i )) is an image of the unique p ath P ( v i , v ′ i ) in T whih onne ts v i and v ′ i . Pr o of. W e pro v e lemma b y an indution on the n um b er of elemen ts of the set V ∗ . Denote b y ♯A a n um b er of elemen ts of a set A . Let us remark that ♯V ∗ ≥ 2 sine V ter ⊆ V ∗ and ♯V ter ≥ 2 for a non degenerated tree T (it is easily v eried b y indution on the n um b er of v erties). Base of indution. Let ♯V ∗ = 2 . F rom what w as said ab o v e it follo ws that V ∗ = V ter . So, a tree satises a ondition ♯V ter = 2 . F or su h trees it is easy to pro v e b y indution on the n um b er of v erties that ev ery v ertex of V \ V ter has degree 2. In other w ords it is adjaen t to t w o edges. If a tree is onsidered as CW-omplex (i.e. 0-ells are its v erties and 1-ells are its edges), then a top ologial spae T is homeomorphi to a segmen t with a set of the endp oin ts whi h oinides with V ∗ = V ter . Let ϕ ( T ) b e a ut of a disk D 2 b et w een ϕ ( v 1 ) and ϕ ( v 2 ) , where { v 1 , v 2 } = V ter . Let us x a homeomorphism Φ 0 : ∂ D 2 ∪ ϕ ( T ) → ∂ D 2 ∪ ([ − 1 , 1] × { 0 } ) , 3 su h that Φ 0 ◦ ϕ ( v 1 ) = ( − 1 , 0) , Φ 0 ◦ ϕ ( v 2 ) = (1 , 0) , Φ 0 ( ∂ D 2 ) = ∂ D 2 , Φ 0 ◦ ϕ ( T ) = [ − 1 , 1] × { 0 } . By Sherni's theorem [11, 13℄ w e an nd a homeomorphism Φ : R 2 → R 2 whi h extends Φ 0 . It is ob vious that an em b edding Φ ◦ ϕ : T → R 2 omplies with the onditions of lemma. F rom fat that Φ is homeomorphism it follo ws that ϕ satises onditions of lemma. Step of indution. Supp ose that for some n > 2 lemma is pro v ed for all trees with ♯V ∗ < n and their em b eddings in to R 2 whi h hold Conditions ( 1) and (2 ). Let a tree T su h that V ter ⊆ V ∗ , ♯V ∗ = n , and an em b edding ϕ : T → R 2 whi h satisfy Conditions (2) is xed. As w e notied ab o v e the set V ter on tains at least t w o elemen ts w 1 , w 2 ∈ V ter . Let us onsider the path P ( w 1 , w 2 ) whi h onnets those v erties. Supp ose that it passes through the v erties in the follo wing order w 1 = u 0 , u 1 , . . . , u k − 1 , u k = w 2 . Ev ery v ertex u 1 , . . . , u k − 1 has degree at least 2 sine it is adjaen t to t w o edges of P ( w 1 , w 2 ) . There exists a v ertex u s , s ∈ { 1 , . . . , k − 1 } su h that (i) a degree of u i equals to 2 and u i / ∈ V ∗ for i ∈ { 1 , . . . , s − 1 } ; (ii) either a degree of u s is greater than 2 or u s ∈ V ∗ and a degree of u s equals to 2. Remark that a degree of u s do es not equal to 1. Otherwise, the orrelations u s = w 2 , T = P ( v 1 , v 2 ) , V ∗ = { w 1 , w 2 } , ♯V ∗ = 2 should b e satised but w e assumed that ♯V ∗ ≥ 3 . Let us onsider a path P ( w 1 , u s ) = P ( u 0 , u s ) . Supp ose that it passes through edges e 1 , . . . , e s suessiv ely . W e onsider a subgraph T ′ of T with the set of v erties and edges, resp etiv ely , as follo w es V ( T ′ ) = V \ { u 0 , . . . , u s − 1 } , E ( T ′ ) = E \ { e 1 , . . . , e s } . By onstrution u 0 ∈ V ter ( T ) and u 0 is adjaen t to e 1 in T ; ev ery v ertex u i , i ∈ { 1 , . . . , s − 1 } has degree 2 th us it is adjaen t only to e i and e i +1 in T . Therefore a graph T ′ is dened orretly . A graph T ′ has no yles sine it is a subgraph of T . Let us v erify that T ′ is onneted. Let v ′ , v ′′ ∈ V ( T ′ ) and P ( v ′ , v ′′ ) b e a path whi h onnets v erties v ′ and v ′′ in T . Then a path P ( v ′ , v ′′ ) do es not pass through a v ertex u 0 = w 1 sine u 0 ∈ V ter and only one edge e 1 is adjaen t to this v ertex. Th us e 1 / ∈ P ( v ′ , v ′′ ) . Similarly , if s ≥ 2 then e 2 / ∈ P ( v ′ , v ′′ ) sine an edge e 2 is adjaen t to a v ertex u 1 whi h is in addition adjaen t only to e 1 and e 1 / ∈ P ( v ′ , v ′′ ) . Similarly , b y indution w e pro v e that e i / ∈ P ( v ′ , v ′′ ) for ev ery i ∈ { 1 , . . . , s } . Th us a path P ( v ′ , v ′′ ) onnets v erties v ′ and v ′′ in T ′ . Therefore a graph T ′ is onneted. W e v eried that T ′ is a tree. Let us dene V ∗ ( T ′ ) = V ∗ ( T ) ∩ V ( T ′ ) , ϕ 0 = ϕ | T ′ : T ′ → R 2 . By denition of a set V ∗ ( T ′ ) it is ob vious that a map ϕ 0 satises 4 ondition (2). Also ♯V ∗ ( T ′ ) < ♯V ∗ ( T ) sine u 0 ∈ V ∗ ( T ) \ V ∗ ( T ′ ) . Th us ♯V ∗ ( T ′ ) < n . Let us he k that V ter ( T ′ ) ⊆ V ∗ ( T ′ ) . By onstrution for ev ery v ertex v 6 = u s of T ′ its degrees oinide in T and T ′ . The degree of u s in T ′ is on one less then degree of u s in T . Th us V ter ( T ′ ) ⊆ V ter ( T ) ∪ { u s } . If u s ∈ V ∗ ( T ) , then V ter ( T ′ ) ⊆ V ter ( T ) ∪ V ∗ ( T ) ⊆ V ∗ ( T ) . Therefore V ter ( T ′ ) ⊆ V ∗ ( T ) ∩ V ( T ′ ) = V ∗ ( T ′ ) . Let u s / ∈ V ∗ ( T ) . By denition the degree of u s in T is not less then 3 and a degree of u s in T ′ is not less then 2. Th us V ter ( T ′ ) ⊆ V ter ( T ) ⊆ V ∗ ( T ) . So, as ab o v e, V ter ( T ′ ) ⊆ V ∗ ( T ′ ) . By indution lemma holds true for a tree T ′ and an em b edding ϕ 0 : T ′ → R 2 . Denote b y W 0 = R 2 \ D 2 , W 1 , . . . , W r onneted omp onen ts of a set R 2 \ ( ϕ 0 ( T ′ ) ∪ ∂ D 2 ) . It is ob vious that ϕ ( T ) = ϕ ( T ′ ) ∪ ϕ ( P ( u 0 , u s )) = ϕ 0 ( T ′ ) ∪ ϕ ( P ( u 0 , u s )) . Therefore ϕ ( T ) ∪ ∂ D 2 = ( ϕ 0 ( T ′ ) ∪ ∂ D 2 ) ∪ ϕ ( P ( u 0 , u s )) . By onstrution w e get that ( ϕ ( T ′ ) ∪ ∂ D 2 ) ∩ ϕ ( P ( u 0 , u s )) = { ϕ ( u 0 ) , ϕ ( u s ) } . Denote J = ϕ ( P ( u 0 , u s )) . The set J 0 = J \ { ϕ ( u 0 ) , ϕ ( u s ) } is a homeomorphi image of in terv al th us it is onneted. But b esides J 0 ∩ ( ϕ 0 ( T ′ ) ∪ ∂ D 2 ) = ∅ th us there exists a omp onen t W j whi h on tains J 0 (it is easy to see that j 6 = 0 ). By assumption of indution the b oundary of disk W j is a simple losed urv e ∂ W j = K j ∪ ϕ 0 ( P ( v j , v ′ j )) whi h onsists of an ar K j of a irle ∂ D 2 with the ends ϕ 0 ( v j ) and ϕ 0 ( v ′ j ) and an image of path P ( v j , v ′ j ) whi h onnets v erties v j , v ′ j ∈ V ∗ ( T ′ ) in T ′ (this path also onnets v erties v j and v ′ j in T ). The set J is a homeomorphi image of segmen t and also J 0 ⊆ W j , ϕ ( u 0 ) ∈ ∂ D 2 ⊆ ( R 2 \ W j ) , ϕ ( u s ) ∈ ϕ 0 ( T ′ ) ⊆ ( R 2 \ W j ) . Therefore J is a ut of disk W j b et w een p oin ts ϕ ( v j ) and ϕ ( v ′ j ) . Correlations ϕ ( u s ) ∈ ϕ ( P ( v j , v ′ j )) , ϕ ( u 0 ) ∈ K j \ { ϕ ( v j ) , ϕ ( v ′ j ) } = ∂ W j \ ϕ ( T ′ ) hold true sine u 0 / ∈ V ( T ′ ) and ϕ ( u 0 ) / ∈ ϕ ( T ′ ) . So, a set W j \ ( ∂ W j ∪ ϕ ( P ( u 0 , u s ))) has t w o onneted omp onen ts W 1 j , W 2 j whi h are homeomorphi to op en disks and b ounded b y simple losed urv es. W e remark that the ar ϕ ( P ( v j , v ′ j )) is not a p oin t, otherwise the orrelations K j ∼ = ∂ D 2 , ϕ 0 ( T ′ ) ∩ ∂ D 2 = { ϕ ( v j ) = ϕ ( v ′ j ) } , ♯V ∗ ( T ′ ) = ♯ ( ϕ 0 ( T ′ ) ∩ ∂ D 2 ) = 1 should hold true. Th us p oin ts ϕ ( v j ) and ϕ ( v ′ j ) are dieren t. F rom the inlusions ϕ ( u s ) ∈ ϕ ( P ( v j , v ′ j )) , ϕ ( u 0 ) ∈ ∂ W j \ ϕ ( P ( v j , v ′ j )) it follo ws that p oin ts ϕ ( v j ) and ϕ ( v ′ j ) an not b e on tained in a set ∂ W 1 j ∩ ∂ W 2 j = ϕ ( P ( u 0 , u s )) sim ultaneously . Let ϕ ( v j ) ∈ ∂ W 1 j , ϕ ( v ′ j ) ∈ ∂ W 2 j . By those orrelations the sets W 1 j and W 2 j are dened uniquely . P oin ts ϕ ( u 0 ) , ϕ ( u s ) divide the irle on to t w o ars R 1 , R 2 with R 1 ⊆ ∂ W 1 j \ W 2 j , R 2 ⊆ ∂ W 2 j \ W 1 j . 5 Supp ose for some edge e ∈ E ( T ) its image is on tained in ∂ W j . Then the image of e without the ends is onneted set and b elongs to ∂ W j \ { ϕ ( u 0 ) , ϕ ( u s ) } = R 1 ∪ R 2 . Th us the image of e without the endp oin ts b elongs to either R 1 or R 2 . The path whi h onnets v erties v j and v ′ j in T ′ passes through the v erties v j = ˆ v 0 , ˆ v 1 , . . . , ˆ v k = v ′ j and through the edges ˆ e 1 , . . . , ˆ e k in this order. If ϕ ( ˆ v i ) ∈ R 1 for some i ∈ { 0 , . . . , k } , then ϕ ( ˆ v i ) ∈ R 2 \ R 2 and ( ϕ ( ˆ e i ) \ { ϕ ( ˆ v i ) , ϕ ( ˆ v i +1 ) } ) ∩ ( R 2 \ R 2 ) 6 = ∅ sine a p oin t ϕ ( ˆ v i ) is a b oundary for the set ϕ ( ˆ e i ) \ { ϕ ( ˆ v i ) , ϕ ( ˆ v i +1 ) } but R 2 \ R 2 is an op en neigh b orho o d of this p oin t. F rom what w e said it follo ws that ϕ ( ˆ e i ) \ ϕ ( ˆ v i +1 ) ⊆ R 1 . Therefore ϕ ( ˆ v i +1 ) ∈ R 1 = R 1 ∪ { ϕ ( u 0 ) , ϕ ( u s ) } . Indeed, either ϕ ( ˆ v i +1 ) ∈ R 1 or ϕ ( ˆ v i +1 ) = ϕ ( u s ) (and ˆ v i +1 = u s ) sine u 0 / ∈ V ( T ′ ) b y onstrution. By assumption of indution ϕ ( u s ) ∈ ϕ ( P ( v j , v ′ j )) = ϕ ( ˆ v 0 , ˆ v k )) . Therefore u s ∈ { ˆ v 0 , . . . , ˆ v k } and there exists an index k 0 ∈ { 0 , . . . , k } su h that u s = ˆ v k 0 . The indutiv e appliation of our previous argumen t leads us to orrelations ϕ ( P ( ˆ v 0 , u s )) \ ϕ ( u s ) = ϕ ( P ( v j , u s )) \ ϕ ( u s ) ⊆ R 1 (in the ase when v j = u s w e get ϕ ( P ( v j , u s )) = ϕ ( u s ) ). Similar argumen t giv e ϕ ( P ( u s , v ′ j )) \ ϕ ( u s ) ⊆ R 2 . Finally w e get ∂ W 1 j = R 1 ∪ ϕ ( P ( u 0 , u s )) = R 1 ∪ J , ∂ W 2 j = R 2 ∪ J ; ∂ W 1 j ∩ ϕ ( P ( v j , v ′ j )) = ∂ W 1 j ∩ ( ϕ ( P ( v j , u s )) ∪ ϕ ( P ( u s , v ′ j ))) = ϕ ( P ( v j , u s )) ; ∂ W 2 j ∩ ϕ ( P ( v j , v ′ j )) = ϕ ( P ( u s , v ′ j )) . Therefore ϕ ( T ) ∩ ∂ W 1 j = ( ϕ ( T ′ ) ∪ J ) ∩ ∂ W 1 j = ( ϕ ( P ( v j , v ′ j )) ∪ J ) ∩ ∂ W 1 j = ϕ ( P ( v j , u s )) ∪ ϕ ( P ( u 0 , u s )) = ϕ ( P ( v j , u 0 )) ; ϕ ( T ) ∩ ∂ W 2 j = ϕ ( P ( v ′ j , u 0 )) . It is easy to see that ϕ ( v j ) 6 = ϕ ( u 0 ) and ϕ ( v ′ j ) 6 = ϕ ( u 0 ) sine v j , v ′ j ∈ V ( T ′ ) but u 0 / ∈ V ( T ′ ) . Hene a set ϕ ( P ( v j , u 0 )) \ { ϕ ( v j ) , ϕ ( u 0 ) } is one of t w o onneted omp onen ts of the set ∂ W 1 j \ { ϕ ( v j ) , ϕ ( u 0 ) } . Another onneted omp onen t of this set is on tained in ∂ W j \ ϕ ( T ′ ) = K j ⊆ ∂ D 2 th us it is an ar of irle ∂ D 2 whi h onnets p oin ts ϕ ( v j ) and ϕ ( u 0 ) . Denote it b y K 1 j . Similarly , ∂ W 2 j = ϕ ( P ( v ′ j , u 0 )) ∪ K 2 j , where K 2 j is an ar of ∂ D 2 whi h onnets p oin ts ϕ ( v ′ j ) and ϕ ( u 0 ) . W e pro v ed that a omplimen t R 2 \ ( ϕ ( T ) ∪ ∂ D 2 ) has a nite n um b er of onneted omp onen ts R 2 \ D 2 = W 0 , W 1 , . . . , W j − 1 , W 1 j , W 2 j , W j +1 , . . . , W r ; and the omp onen ts W 1 j and W 2 j satisfy the onditions of lemma. Finally w e remark that the orrelations ∂ W k ∩ ϕ ( T ) = ∂ W k ∩ ϕ ( T ′ ) = ∂ W k ∩ ϕ 0 ( T ′ ) hold true for k > 0 , k 6 = j th us ∂ W k = K k ∪ ϕ 0 ( P ( v k , v ′ k )) = K k ∪ ϕ ( P ( v k , v ′ k )) and the omp onen t W k satises lemma. 6 Corollary 1.1. L et T b e a tr e e with xe d subset of verti es V ∗ ⊇ V ter and ϕ : T → R 2 an emb e dding whih satises (2) . Then the fol lowing onditions hold true. 1)In notation of L emma 1.1 L i ∩ ϕ ( T ) = { ϕ ( v i ) , ϕ ( v ′ i ) } , i = 1 , . . . , m . 2) If ther e exists an ar L of ir le ∂ D 2 with the ends ϕ ( u 1 ) , ϕ ( u 2 ) suh that L ∩ ϕ ( T ) = { ϕ ( u 1 ) , ϕ ( u 2 ) } for some u 1 , u 2 ∈ V ∗ , then ther e exists k ∈ { 1 , . . . , m } suh that L ∪ ϕ ( P ( u 1 , u 2 )) = ∂ U k (then L = L k , u 1 = v k , u 2 = v ′ k ). Pr o of. 1) Supp ose that an ar L i \ { ϕ ( v i ) , ϕ ( v ′ i ) } on tains a p oin t ϕ ( v ) ∈ ϕ ( T ) for some i ∈ { 1 , . . . , m } . Th us v ∈ V ∗ . Let e ∈ E ( T ) b e an edge of graph T whi h is adjaen t to a v ertex v and v ′ ∈ V b e another end of the edge e . A set J 0 = ϕ ( e ) \ { ϕ ( v ) , ϕ ( v ′ ) } is onneted, ϕ ( v ) is a b oundary p oin t of it, W = R 2 \ ϕ ( P ( v i , v ′ i )) is an op en neigh b orho o d of a p oin t ϕ ( v ) . Th us J 0 ∩ W 6 = ∅ and e / ∈ P ( v i , v ′ i ) . Hene J 0 ∩ ϕ ( P ( v i , v ′ i )) = ∅ . By the onditions of lemma also J 0 ∩ ∂ D 2 = ∅ . A set ϕ ( P ( v i , v ′ i )) is a ut of losed disk D 2 . Ob viously , b y onstrution a set Q = U i \ ϕ ( P ( v i , v ′ i )) = U i ∪ ( L i \ { ϕ ( v i ) , ϕ ( v ′ i ) } ) is a onneted omp onen t of the omplimen t D 2 \ ϕ ( P ( v i , v ′ i )) whi h on tains a p oin t ϕ ( v ) . That p oin t is a b oundary p oin t of the onneted subset J 0 of a spae D 2 \ ϕ ( P ( v i , v ′ i )) therefore J 0 ⊆ Q . But U i ⊆ R 2 \ ϕ ( T ) , L i ⊆ ∂ D 2 and J 0 ⊆ ϕ ( T ) \ ϕ ( V ) ⊆ ϕ ( T ) \ ∂ D 2 . Th us J 0 ∩ Q ⊆ ( J 0 ∩ U i ) ∪ ( J 0 ∩ L i ) = ∅ . The on tradition obtained is a last step of the pro of of rst ondition of orol- lary . 2) Supp ort that there exists an ar L of ∂ D 2 with the ends in p oin ts ϕ ( u 1 ) , ϕ ( u 2 ) su h that L ∩ ϕ ( T ) = { ϕ ( u 1 ) , ϕ ( u 2 ) } for some u 1 , u 2 ∈ V ∗ . An ar L b ounders to some onneted omp onen t U k , k ≥ 1 of the ompli- men t R 2 \ ( ϕ ( T ) ∪ ∂ D 2 ) . F rom Lemma 1.1 and rst ondition of orollary it follo ws that { u 1 , u 2 } = { v k , v ′ k } . Th us v erties u 1 and u 2 an b e onneted b y a path ˜ P ( u 1 , u 2 ) = P ( v k , v k ′ ) whi h satises Lemma 1.1 . A graph T is a tree th us P ( u 1 , u 2 ) = ˜ P ( u 1 , u 2 ) = P ( v k , v k ′ ) . Let T b e a tree with a xed subset of v erties V ∗ and ϕ : T → R 2 is an em b edding whi h satisfy ( 1 ) and (2). The pair of v erties v 1 , v 2 ∈ V ∗ , v 1 6 = v 2 is said to b e adja ent on a ir le ∂ D 2 if there exists an ar L of this irle with the ends ϕ ( v 1 ) and ϕ ( v 2 ) su h that L ∩ ϕ ( T ) = { ϕ ( v 1 ) , ϕ ( v 2 ) } holds true for it. Denote b y P a set of all paths in T whi h onnet adjaen t pairs of v erties. 7 Corollary 1.2. If ♯V ∗ ≥ 3 , then a orr esp onden e Θ : { U 1 , . . . , U m } → P , Θ( U i ) = P ( v i , v ′ i ) , is a bije tive map. Pr o of. It is suien t to he k an injetivit y of the map Θ . Supp ose that the follo wing equalities hold true ∂ U i = L i ∪ ϕ ( P ( v , v ′ )) , ∂ U j = L j ∪ ϕ ( P ( v , v ′ )) for some i , j ∈ { 1 , . . . , m } , i 6 = j . Then L i ∩ L j = { ϕ ( v ) , ϕ ( v ′ ) } , L i ∪ L j ∼ = S 1 therefore L i ∪ L j = ∂ D 2 . But from Corollary 1.1 it follo ws that ∅ = ( L i ∪ L j \ { ϕ ( v ) , ϕ ( v ′ ) } ) ∩ ϕ ( T ) . Therefore ♯ ( ∂ D 2 ∩ ϕ ( T )) = ♯V ∗ ≤ 2 and it on tradits to the onditions of orollary . 1.2. On relations dened on nite sets. A t rst w e remind that a ternary relation O on the set A is an y subset of the 3 r d artesian p o w er A 3 : O ⊆ A 3 . Let A b e a set, O a ternary relation on A whi h is asymmetri ( ( x, y , z ) ∈ O ⇒ ( z , y , x ) ∈ O ), transitiv e ( x, y , z ) ∈ O , ( x, z , u ) ∈ O ⇒ ( x, y , u ∈ O ) and yli ( x, y , z ) ∈ O ⇒ ( y , z , x ) ∈ O . Then O is alled a yli order on the set A [12℄. A yli order O is a omplete on a nite set A , ♯A ≥ 3 , if x, y , z ∈ A, x 6 = y 6 = z 6 = x ⇒ there exists a p erm utation ( u, v , w ) of sequene ( x, y , z ) su h that ( u, v , w ) ∈ O . Prop osition 1.1. L et ther e is a omplete yli or der O on some nite set A , ♯A ≥ 3 . Then for every a ∈ A ther e exist unique a ′ , a ′′ ∈ A suh that • O ( a ′ , a, b ) for al l b ∈ A \ { a, a ′ } ; • O ( a, a ′′ , b ) for al l b ∈ A \ { a, a ′′ } , and a ′ 6 = a ′′ . Pr o of. Let us x a ∈ A . By using [12℄ w e an onstrut a binary relation ρ up to the relation O with the help of the follo wing ondition O ( a, a 1 , a 2 ) ⇔ a 1 ρ a 2 . It is easy to v erify that the relation ρ denes a strit linear order on a set A \ { a } . The set A \ { a } is nite therefore there exist a minimal elemen t a ′ and maximal elemen t a ′′ with resp et to the order ρ on this set. It is ob vious that they satisfy onditions of prop osition b y denition. Finally , a ′ 6 = a ′′ sine ♯ ( A \ { a } ) ≥ 2 . Denition 1.1. L et ther e is a omplete yli or der O on a set A , ♯A ≥ 3 . Elements a 1 , a 2 ∈ A ar e said to b e adjaen t with r esp e t to a yli or der O if one of the fol lowing onditions holds: • O ( a 1 , a 2 , b ) for al l b ∈ A \ { a 1 , a 2 } ; 8 • O ( a 2 , a 1 , b ) for al l b ∈ A \ { a 1 , a 2 } . Remark 1.1. F r om Pr op osition 1.1 it fol lows that every element has exatly two adja ent elements on a nite set A with a omplete yli or der. Denition 1.2. L et A b e a nite set. A binary r elation ρ on A is said to b e on v enien t if 1) for al l a , b ∈ A fr om aρb it fol lows that a 6 = b ; 2) for every a ∈ A ther e is no mor e than one a ′ ∈ A suh that aρa ′ ; 3) for every a ∈ A ther e is no mor e than one a ′′ ∈ A suh that a ′′ ρa . W e remind that a gr aph of the relation ρ on A is a set { ( a, b ) ∈ A × A | aρb } . Let ρ b e a on v enien t relation on a nite set A , ˆ ρ b e a minimal relation of equiv alene whi h on tains ρ . Let us remind that a graph ˆ ρ onsists of • all pairs ( a, b ) su h that there exist k = k ( a, b ) ∈ N and a sequene a = a 0 , a 1 , . . . , a k = b whi h omply with one of the follo wing onditions a i − 1 ρa i , a i ρa i − 1 for ev ery i ∈ { 1 , . . . , k } ; • pairs ( a, a ) , a ∈ A . W e distinguished a diagonal ∆ A × A sine, in general, there ould exist a ∈ A su h that neither aρb nor bρa holds true for all b ∈ A . The relation ˆ ρ generates a partition f of A on to lasses of equiv alene. Prop osition 1.2. L et B ∈ f b e a lass of e quivalen e of the r elation ˆ ρ . Then ther e exists no mor e than one element b ∈ B whih is in the r elation ρ with no element of A . Pr o of. W e remark that if either aρb or bρa and b ∈ B , then a ∈ B b y denition of B . It is ob vious that if ♯B = 1 then prop osition holds true. Let ♯B ≥ 2 . Let a 0 , a 1 , . . . , a k b e a xed sequene of pairwise dieren t elemen ts of B su h that the orrelation a i − 1 ρa i holds true for an y i ∈ { 1 , . . . , k } . If there exists b ∈ B \ { a 0 , . . . , a k } , then there exists b ′ ∈ B \ { a 0 , . . . , a k } su h that either b ′ ρa 0 or a k ρb ′ . Let us v erify it. By denition of a set B there exists a sequene b = c 0 , c 1 , . . . , c m = a 0 su h that either c j − 1 ρc j or c j ρc j − 1 holds true for all j ∈ { 1 , . . . , m } . F rom orrelations c 0 / ∈ { a 0 , . . . , a k } and c m ∈ { a 0 , . . . , a k } it follo ws that there is s ∈ { 0 , . . . , m } su h that c s − 1 / ∈ { a 0 , . . . , a k } but c s ∈ { a 0 , . . . , a k } . Th us c s = a r for some r ∈ { 0 , . . . , k } . Let c s − 1 ρc s , i.e. c s − 1 ρa r . Then r = 0 . Really , if r ≥ 1 , then a r − 1 ρa r . By onstrution c r − 1 6 = a r − 1 therefore a orrelation c s − 1 ρa r on tradits to ondition 3) of Denition 1.2 . Similarly , if c s ρc s − 1 , then c s = a k . It is easy to see that elemen t b ′ = c s − 1 satises onditions of prop osition. 9 F rom what w e said ab o v e it follo ws that if for some pairwise dieren t a 0 , . . . , a k ∈ B inequalit y { a 0 , . . . , a k } 6 = B and relation a i − 1 ρa i , i ∈ { 1 , . . . , k } hold true, then there are pairwise dieren t a ′ 0 , . . . , a ′ k +1 ∈ B su h that a ′ i − 1 ρa ′ i , i ∈ { 1 , . . . , k + 1 } hold true for them. By denition the set B on tains t w o elemen ts b ′ , b ′′ ∈ B su h that b ′ ρb ′′ . So, b y a nite n um b er of steps (the set B is nite) w e an index all elemen ts of B in su h w a y that the follo wing orrelations hold true a i − 1 ρa i , i ∈ { 1 , . . . , n } ; (3) { a 0 , . . . , a n } = B . Therefore only elemen t a n ∈ B an satisfy onditions of the prop osition. Let µ b e some relation on a set A . Denition 1.3. Elements b 0 , . . . , b n ∈ A , n ≥ 1 ar e said to gener ate µ -yle if a gr aph of the r elation µ ontains a set (4) { ( b 0 , b 1 ) , . . . , ( b n − 1 , b n ) , ( b n , b 0 ) } . Denition 1.4. Elements b 0 , . . . , b n ∈ A , n ≥ 0 to gener ate µ - hain if for arbi- tr ary a ∈ A the p airs ( a, b 0 ) and ( b n , a ) do not b elong to a gr aph of µ and for n ≥ 1 a gr aph of the r elation µ ontains a set (5) { ( b 0 , b 1 ) , . . . , ( b n − 1 , b n ) } . Corollary 1.3. L et ρ b e a onvenient r elation, B ∈ f a lass of e quivalen e of the r elation ˆ ρ . Then the elements of B gener ate either ρ -yle or ρ -hain. In the rst ase a gr aph of the r estrition of ρ on the set B is of form ( 4 ) and in the other it has form (5) . Pr o of. Let us order the elemen ts of B in su h w a y that (3) holds true for them. If there exists a ∈ A su h that a n ρa , then a ∈ B . Conditions 1) and 3) of Denition 1.2 obstrut to hold orrelation a i ρa j for i 6 = j − 1 , j ∈ { 1 , . . . , n } . Therefore a = a 0 and a n ρa 0 . Similarly , if there exists a ∈ A whi h aρa 0 , then from onditions 1) and 2) of Denition 1.2 it follo ws that a = a n and a n ρa 0 . So, either a orrelation a n ρa 0 holds true or for ev ery a ∈ A neither aρa 0 nor a n ρa holds true. In the rst ase the elemen ts of B generate ρ -yle (if a n ρa 0 , then a n 6 = a 0 and ♯B ≥ 2 b y denition), in the other ase w e get ρ - hain. Corollary 1.4. L et the elements of B ⊆ A gener ate either ρ -yle or ρ -hain. Then B is a lass of e quivalen e of the r elation ˆ ρ . If the elements of B ⊆ A gener ate ρ -hain, then the r elation ρ gener ates a ful l line ar or der on B . Pr o of. Let ˆ ρ b e a minimal relation of equiv alene whi h on tains ρ . By denition the set B b elongs to the unique lass of equiv alene of the relation ˆ ρ . Denote it b y ˆ B . 10 By denition the set B satises (3). If there exists b ∈ ˆ B \ B , then, as w e v eried in the pro of of Prop osition 1.2, there is b ′ ∈ ˆ B \ B su h that (6) b ′ ρa 0 or a n ρb ′ . This on tradits to denition of ρ - hain. If the elemen ts of B generate ρ -yle, then it follo ws from the denition of on v enien t relation that (7) a n ρa 0 , see Corollary 1.3. By using onditions 2) and 3) of a on v enien t relation from equalit y b ′ / ∈ B w e an onlude that (6) and (7) an not b e satised sim ultane- ously . So, a set B is a lass of equiv alene of the relation ˆ ρ . If elemen ts of the set B generate a hain, then a graph of a restrition of the relation ρ on B has form (5), see Corollary 1.3 . Therefore ρ generates a linear order on the set B . Denition 1.5. L et O b e a omplete yli or der on A , ♯A ≥ 3 . O is said to indu e a binary r elation ρ O on a A a or ding to the fol lowing rule: aρ O b if O ( a, b, c ) ∀ c ∈ A \ { a, b } . F rom Prop osition 1.1 it follo ws that a relation ρ O is on v enien t. Prop osition 1.3. If O is a omplete yli or der on A , then al l elements of A gener ate ρ O -yle. Pr o of. Let ˆ ρ O b e a minimal relation of equiv alene whi h on tains ρ O . F rom Prop osition 1.1 and Corollaries 1.3 and 1.4 it follo ws that ev ery lass of equiv alene of the relation ˆ ρ O is ρ O -yle and there are no an y other ρ O -yles. Let B = { b 0 , . . . , b k } b e some lass of equiv alene of the relation ˆ ρ O and the follo wing orrelations are satised b 0 ρb 1 , . . . , b k − 1 ρb k , b k ρb 0 . Supp ort that B A . Let us x a ∈ A \ B . By denition the follo wing orrelations hold true O ( b i − 1 , b i , a ) , i ∈ { 1 , . . . , k } ; O ( b k , b 0 , a ) . Th us it follo ws from denition of yli order it follo ws that O ( a, b i − 1 , b i ) , i ∈ { 1 , . . . , k } ; O ( a, b k , b 0 ) . F rom denition it also follo ws that if O ( a, b 0 , b i − 1 ) and O ( a, b i − 1 , b i ) , then O ( a, b 0 , b i ) . Therefore starting from O ( a, b 0 , b 1 ) in the nite n um b er of steps w e get O ( a, b 0 , b k ) . 11 Th us O ( a, b k , b 0 ) and O ( a, b 0 , b k ) should b e satised sim ultaneously but it on- tradits to an tisymmetry of yli order. Therefore all elemen ts of a set A are equiv alen t under ˆ ρ O and generate ρ O - yle. Denition 1.6. L et ρ b e a onvenient r elation on a nite set A . W e dene a ternary r elation O ρ on A with the help of the fol lowing rule. The or der e d triple ( a 1 , a 2 , a 3 ) of A is said to b e in the r elation O ρ if a 1 6 = a 2 6 = a 3 6 = a 1 and ther e ar e (8) a 1 = a 12 0 , a 12 1 , . . . , a 12 m (1) = a 2 = a 23 0 , . . . , a 23 m (2) = a 3 = a 31 0 , . . . , a 31 m (3) = a 1 , whih satisfy the fol lowing onditions: • a sr n − 1 ρa sr n for al l n ∈ { 1 , . . . , m ( s ) } and ( s + 1 ) ≡ r (mo d 3) ; • a sr n / ∈ { a 1 , a 2 , a 3 } for al l n ∈ { 1 , . . . , m ( s ) − 1 } and ( s + 1 ) ≡ r (mo d 3) . Prop osition 1.4. The r elation O ρ is a yli or der on A . Pr o of. F rom denition it is ob vious that the relation O ρ is yli. Let us remark that from denition if O ρ ( a 1 , a 2 , a 3 ) , then all elemen ts of a set (8) (in partiular elemen ts a 1 , a 2 and a 3 ) b elong to the same lass of equiv alene of minimal equiv alene relation ˆ ρ whi h on tains ρ . W e should v erify that all elemen ts a sr n , n ∈ { 1 , . . . , m ( s ) } , ( s + 1) ≡ r (mo d 3) are dieren t. Supp ose that it is not true and there are t w o dieren t sets of indexes su h that a sr n = a tτ k , n ∈ { 1 , . . . , m ( s ) } , k ∈ { 1 , . . . , m ( t ) } , ( s + 1) ≡ r (mo d 3) , ( t + 1) ≡ τ (mo d 3) . Let us onsider t w o sequenes ( b 1 , . . . , b i ) = ( a sr n , a sr n +1 , . . . , a sr m ( s ) , . . . , a tτ 0 , a tτ 1 , . . . , a tτ k − 1 , a tτ k ) , ( c 1 , . . . , c j ) = ( a tτ k , a tτ k +1 , . . . , a tτ m ( t ) , . . . , a sr 0 , a sr 1 , . . . , a sr n − 1 , a sr n ) . Those t w o sequenes satisfy the follo wing onditions: • b l − 1 ρb l for all l ∈ { 1 , . . . , i } ; • c l − 1 ρc l for all l ∈ { 1 , . . . , j } ; • b i = c 1 = c j = b 1 ; • there exists ˆ a ∈ { a 1 , a 2 , a 3 } su h that either ˆ a ∈ { b 1 , . . . , b i } \ { c 1 , . . . , c j } or ˆ a ∈ { c 1 , . . . , c j } \ { b 1 , . . . , b i } sine ev ery elemen t a 1 , a 2 , a 3 is on tained exatly one in the sequene (8) b y denition. Let ˆ a / ∈ { b 1 , . . . , b i } . By denition the elemen ts b 1 , . . . , b i generate a yle there- fore the set { b 1 , . . . , b i } is a lass of equiv alene of the relation ˆ ρ , see Corollary 1.4. But it on tradits to the ondition that all elemen ts of the set ( 8) b elong to the same lass of equiv alene of the relation ˆ ρ . The ase when ˆ a / ∈ { c 1 , . . . , c j } an b e onsidered similarly . Therefore all elemen ts of the set (8) are dieren t. 12 Let O ρ ( a 1 , a 2 , a 3 ) and O ρ ( a 3 , a 2 , a 1 ) hold true sim ultaneously . Then from deni- tion there are t w o sequenes a 3 = a 31 0 , a 31 1 , . . . , a 31 m (3) = a 1 and a 1 = b 31 0 , b 31 1 , . . . , b 31 n (3) = a 3 su h that • a 31 i − 1 ρa 31 i for all i ∈ { 1 , . . . , m (3) } ; • b 31 j − 1 ρb 31 j for all j ∈ { 1 , . . . , n (3) } ; • a 2 / ∈ { a 31 0 , . . . , a 31 m (3) , b 31 0 , . . . , b 31 n (3) } . It is ob vious that there is k ∈ { 1 , . . . , m (3 ) } su h that a 31 i / ∈ { b 31 0 , . . . , b 31 n (3) } for i < k but a 31 k ∈ { b 31 0 , . . . , b 31 n (3) } . Hene a 31 k = b 31 l for some l ∈ { 1 , . . . , n (3) } and a 31 k ρb 31 l +1 . It is lear that all elemen ts of the follo wing sequene a 3 = a 31 0 , . . . , a 31 k , b 31 l +1 , . . . , b 31 n (3) are dieren t and generate ρ -yle. F urther b y denition a 2 do es not b elong to that sequene. Therefore a 3 = a 31 0 and a 2 b elong to dieren t lasses of equiv alene of relation ˆ ρ , see Corollary 1.4 . On the other hand elemen ts a 1 , a 2 and a 3 m ust b elong to the unique lass of equiv alene ˆ ρ , see ab o v e. This on tradition pro v es the an tisymmetry of the relation O ρ . Let O ρ ( a 1 , a 2 , a 3 ) and O ρ ( a 1 , a 3 , a 4 ) for some a 1 , . . . , a 4 ∈ A . W e should remark that the elemen ts a 1 , . . . , a 4 are pairwise dieren t. Really , b y denition a 1 6 = a 3 and { a 1 , a 3 } ∩ { a 2 , a 4 } = ∅ . If a 2 = a 4 , then from a yliit y of relation O ρ it follo ws that O ρ ( a 3 , a 1 , a 2 ) and O ρ ( a 4 , a 1 , a 3 ) = O ρ ( a 2 , a 1 , a 3 ) . But it is imp ossible sine a relation O ρ is an tisymmetri. Let us onsider a sequene (8). Its elemen ts generate ρ -yle. W e will pro v e that a 4 ∈ { a 31 1 , . . . , a 31 m (3) − 1 } . Supp ose that a 4 ∈ { a 12 1 , . . . , a 12 m (1) − 1 } . Then a 4 = a 12 k , k ∈ { 1 , . . . , m (1) − 1 } . W e onsider the sequenes ( b 12 0 , . . . , b 12 t (1) ) = ( a 1 = a 12 0 , . . . , a 12 k = a 4 ) ; ( b 23 0 , . . . , b 23 t (2) ) = ( a 4 = a 12 k , . . . , a 12 m (1) = a 23 0 , . . . , a 23 m (2) = a 3 ) ; ( b 31 0 , . . . , b 31 t (3) ) = ( a 3 = a 31 0 , . . . , a 31 m (3) = a 1 ) . Join them in to a sequene a 1 = b 12 0 , . . . , b 12 t (1) = a 4 = b 23 0 , . . . , b 23 t (2) = a 3 = b 31 0 , . . . , b 31 t (3) . By the onstrution all elemen ts of su h sequene generate ρ -yle therefore it satises the prop erties whi h are similar to the onditions of the sequene (8). W e get O ρ ( a 1 , a 4 , a 3 ) . Then from a yliit y of the relation O ρ it follo ws that O ρ ( a 4 , a 3 , a 1 ) . But b y the ondition w e ha v e O ρ ( a 1 , a 3 , a 4 ) , moreo v er, w e pro v ed that the relation O ρ is an tisymmetri. Th us the relation O ρ ( a 4 , a 3 , a 1 ) do es not hold true and a 4 / ∈ { a 12 1 , . . . , a 12 m (1) − 1 } . The fat that a 4 / ∈ { a 23 1 , . . . , a 23 m (2) − 1 } an b e pro v ed similarly . Therefore a 4 ∈ { a 31 1 , . . . , a 31 m (3) − 1 } and a 4 = a 31 s for some s ∈ { 1 , . . . , m (3) − 1 } . 13 Let us onsider the sequenes ( c 12 0 , . . . , c 12 τ (1) ) = ( a 1 = a 12 0 , . . . , a 12 m (1) = a 2 ) ; ( c 23 0 , . . . , c 23 τ (2) ) = ( a 2 = a 23 0 , . . . , a 23 m (2) = a 31 0 , . . . , a 31 s = a 4 ) ; ( c 31 0 , . . . , c 31 τ (3) ) = ( a 4 = a 31 s , . . . , a 31 m (3) = a 1 ) . Let us join them in to a sequene a 1 = c 12 0 , . . . , c 12 τ (1) = a 2 = c 23 0 , . . . , c 23 τ (2) = a 4 = c 31 0 , . . . , c 31 τ (3) . By onstrution this sequene satises the onditions of denition 1.6 . Therefore the orrelation O ρ ( a 1 , a 2 , a 4 ) holds true and the relation O ρ is transitiv e. Finally , w e an onlude that the relation O ρ satises all onditions of denition of yli order. Denition 1.7. L et C and D b e yli or ders on sets A and B , r esp e tively. L et ϕ : A → B b e a bije tive map. A map ϕ is al le d a monomorphism of yli or der C into a yli or der D if C ( a 1 , a 2 , a 3 ) ⇒ D ( ϕ ( a 1 ) , ϕ ( a 2 ) , ϕ ( a 3 )) ; it is al le d an epimorphism C onto D if D ( b 1 , b 2 , b 3 ) ⇒ C ( ϕ − 1 ( b 1 ) , ϕ − 1 ( b 2 ) , ϕ − 1 ( b 3 )) ; ϕ is an isomorphism C onto D if C ( a 1 , a 2 , a 3 ) ⇔ D ( ϕ ( a 1 ) , ϕ ( a 2 ) , ϕ ( a 3 )) . Remark 1.2. It is le ar that 1) if ϕ is a monomorphism of yli or der C onto D , then ϕ − 1 is an epimorphism of D onto C ; 2) an isomorphism of the r elations of yli or der is a map whih is a monomor- phism and an epimorphism simultane ously; 3) a r elation of isomorphism is a r elation of e quivalen e. Lemma 1.2. L et C and D b e omplete yli or ders on the sets A and B , r esp e - tively, ϕ : A → B is a bije tive map. If ϕ is either monomorphism or an epimorphism, then ϕ is an isomorphism. Pr o of. Let ϕ b e an epimorphism (in the ase when ϕ is a monomorphism w e onsider a map ϕ − 1 ). Let us he k that ϕ is also a monomorphism. Let C ( a 1 , a 2 , a 3 ) for some a 1 , a 2 , a 3 ∈ A . W e dene b i = ϕ ( a i ) ∈ B , i = 1 , 2 , 3 . F rom denition it follo ws that a 1 6 = a 2 6 = a 3 6 = a 1 . Then b 1 6 = b 2 6 = b 3 6 = b 1 . The yli order D is full therefore there is a p erm utation σ ∈ S (3) su h that D ( b σ (1) , b σ (2) , b σ (3) ) . F rom an epimorphism of ϕ w e an onlude that C ( a σ (1) , a σ (2) , a σ (3) ) . Th us σ is ev en p erm utation. No w from an tisymmetry an yliit y of D it follo ws that D ( b 1 , b 2 , b 3 ) , see [12℄. Therefore w e get D ( ϕ ( a 1 ) , ϕ ( a 2 ) , ϕ ( a 3 )) and ϕ is a monomorphism. Remark 1.3. L emma 1.2 holds true for arbitr ary sets A and B , i.e. they an b e innite. 14 Lemma 1.3. L et O is a r elation of omplete yli or der on the nite set A . Then O = O ρ O , wher e ρ O is a onvenient binary r elation gener ate d by O and O ρ O is a r elation of yli or der gener ate d by the onvenient r elation ρ O . Pr o of. W e should pro v e that the relation O ρ O is full. Let b 1 , b 2 , b 3 b e some pairwise dieren t elemen ts of A . F rom Prop osition 1.3 and Corollary 1.4 the minimal relation of equiv alene ˆ ρ O whi h on tains ρ O has the unique lass of equiv alene B = A . Th us w e an index all elemen ts of A in su h w a y that (3 ) holds true. F rom Corollary 1.3 w e also get a n ρa 0 . It is ob vious that { b 1 , b 2 , b 3 } = { a k 1 , a k 2 , a k 3 } for some 0 ≤ k 1 < k 2 < k 3 ≤ n further there is a in v ersion σ ∈ S (3) su h that a k i = b σ ( i ) , i = 1 , 2 , 3 . Let us onsider the sequenes ( c 12 0 , . . . , c 12 m (1) ) = ( a k 1 , a k 1 +1 , . . . , a k 2 ) ; ( c 23 0 , . . . , c 23 m (2) ) = ( a k 2 , . . . , a k 3 ) ; ( c 31 0 , . . . , c 31 m (3) ) = ( a k 3 , . . . , a n , a 0 , . . . , a k 1 ) . W e an join them in to one a k 1 = c 12 0 , . . . , c 12 m (1) = a k 2 = c 23 0 , . . . , c 23 m (2) = a k 3 = c 31 0 , . . . , c 31 m (3) = a k 1 . By onstrution this sequene satises the onditions of Denition 1.6 th us w e get O ρ O ( a k 1 , a k 2 , a k 3 ) . It means that O ρ O ( b σ (1) , b σ (2) , b σ (3) ) and O ρ O is full. Supp ose that O ρ O ( a 1 , a 2 , a 3 ) holds true for some a 1 , a 2 , a 3 ∈ A . F rom Deni- tion 1.6 it follo ws that there is a sequene a 1 = a 12 0 , . . . , a 12 m (1) = a 2 , su h that a 12 i − 1 ρ O a 12 i for all i ∈ { 1 , . . . , m (1) } . Therefore from denition of the relation ρ O orrelations O ( a 12 i − 1 , a 12 i , a ) follo w for all a ∈ A \ { a 12 i − 1 , a 12 i } , i ∈ { 1 , . . . , m (1) } . In partiular, O ( a 12 i − 1 , a 12 i , a 3 ) , i ∈ { 1 , . . . , m (1) } . F rom yliit y of O it follo ws that the orrelations O ( a 3 , a 12 i − 1 , a 12 i ) , i ∈ { 1 , . . . , m (1) } hold true. Starting from the orrelation O ( a 3 , a 12 0 , a 12 1 ) = O ( a 3 , a 1 , a 12 1 ) , using the pre- vious orrelations and transitivit y of O w e indutiv ely get that O ( a 3 , a 1 , a 12 i ) , i ∈ { 1 , . . . , m (1) } . In partiular, O ( a 3 , a 1 , a 12 m (1) ) = O ( a 3 , a 1 , a 2 ) . F rom a yliit y of O it follo ws that O ( a 1 , a 2 , a 3 ) . Therefore an iden tial map I d A : A → A indues an epimorphism of a omplete yli order O on to a omplete yli order O ρ O . F rom Lemma 1.2 it follo ws that the map I d A is an isomorphism of the yli orders O and O ρ O therefore O = O ρ O . Lemma 1.4. L et ρ b e a onvenient r elation suh that al l elements of a set A , ♯A ≥ 3 gener ate a yle. 15 Supp ose that a gr aph of r elation µ on A is obtaine d fr om a gr aph of ρ by thr owing out two p airs ( b 1 , b ′ 1 ) and ( b 2 , b ′ 2 ) (the ases when either b ′ 1 = b 2 or b ′ 2 = b 1 ar e inlude d). L et ˆ µ b e a minimal r elation of e quivalen e whih ontains µ . Then the r elation µ is onvenient, ˆ µ has exatly two lasses of e quivalen e B 1 and B 2 suh that the elements of e ah of them gener ate µ -hain and the elements b 1 , b 2 ∈ A b elong to the dier ent lasses of e quivalen e of ˆ µ . Pr o of. The fat that µ is a on v enien t relation is trivial orollary from denition. The relation µ do es not on tain yles. In fat, if the elemen ts of some set B ⊆ A generate µ -yle, then elemen ts of B generate ρ -yle. F rom Corollary 1.4 and the ondition of lemma w e get B = A . Then from denition of a yle it follo ws that there is a ∈ A su h that b 1 µa , hene b 1 ρa . But b 1 ρb ′ 1 and b ′ 1 6 = a (b y ondition of lemma b 1 is not in the relation µ with b ′ 1 ). It on tradits to the Condition 2) of denition 1.2 . Th us ev ery lass of equiv alene of the relation ˆ µ is a hain, see Corollary 1.3, and it on tains exatly one elemen t whi h is in the relation µ with no elemen t of A . By ondition of lemma the elemen ts of A generate ρ -yle. F rom Denition 1.2 it follo ws that there is the unique a ′ ∈ A su h that aρa ′ for ev ery a ∈ A . Then aµa ′ , if a / ∈ { b 1 , b 2 } but b 1 and b 2 are the unique elemen ts of the set A whi h are not in the relation µ with an y elemen t of A . no w the statemen t of lemma elemen tary follo ws from what w e said b efore. 1.3. A lo al onnetivit y of t w o dimensional disk in b oundary p oin ts. Denition 1.8. [11, 13℄ L et E b e a subset of a top olo gi al sp a e S and x is some p oint of S ( x do es not ne essarily b elong to E ). A set E is al le d a lo ally onneted in a p oin t x if for every neighb orho o d U of x ther e is a neighb orho o d U ′ ⊆ U of x suh that any two p oints whih b elong to U ′ ∩ E an b e joine d by a onne te d set whih b elongs to U ∩ E . Lemma 1.5. L et D 2 b e a lose d two dimensional disk, x ∈ ∂ D 2 and W an op en neighb orho o d of p oint x in a sp a e D 2 . If for some onne te d omp onents W 1 and W 2 of a set W ∩ ( D 2 \ ∂ D 2 ) the fol lowing orr elation holds true x ∈ W 1 ∩ W 2 , then W 1 = W 2 . Pr o of. Ob viously , w e an assume that D 2 is a standard t w o dimensional disk on a plane. Let U b e a neigh b orho o d of p oin t x in R 2 su h that D 2 ∩ U = W . It is kno wn, see [11, 13℄, that ev ery Jordan domain on the plane is lo ally onneted in all p oin ts of its b oundary . Therefore there exists a neigh b orho o d U ′ of x su h that arbitrary t w o p oin ts whi h b elong to U ′ ∩ ( D 2 \ ∂ D 2 ) an b e onneted b y a onneted set that is on tained in U ∩ ( D 2 \ ∂ D 2 ) . Therefore all p oin ts of the set U ′ ∩ ( D 2 \ ∂ D 2 ) should b elong to the unique onneted omp onen t of a set W ∩ ( D 2 \ ∂ D 2 ) . 16 2. Criterion of a D -planarity of a tree. Let T b e a tree, V a set of its v erties, V ter a set of its terminal v erties and V ∗ ⊆ V a subset of T su h that V ter ⊆ V ∗ . W e assume that if ♯V ∗ ≥ 3 then there is some yli order C dened on V ∗ . Let D 2 = { ( x, y ) ∈ R 2 | x 2 + y 2 ≤ 1 } b e a losed orien ted 2dimensional disk. Denition 2.1. A tr e e T is al le d D -planar if ther e exists an emb e dding ϕ : T → R 2 whih satises (2) and if ♯V ∗ ≥ 3 then a yli or der ϕ ( C ) on ϕ ( V ∗ ) oinides with a yli or der whih is gener ate d by the orientation of ∂ D 2 ∼ = S 1 . Remark 2.1. A map ϕ | V ∗ : V ∗ → ϕ ( V ∗ ) is bije tive when e a ternary r elation ϕ ( C ) on ϕ ( V ∗ ) dene d by fol lowing orr elation C ( v 1 , v 2 , v 3 ) ⇒ ϕ ( C )( ϕ ( v 1 ) , ϕ ( v 2 ) , ϕ ( v 3 )) , v 1 , v 2 , v 3 ∈ V ∗ , is a r elation of yli or der. Remark 2.2. W e an dene a yli or der in a natur al way on an oriente d ir- le S 1 : an or der e d triple of p oints x 1 , x 2 , x 3 ∈ S 1 is yli al ly or der e d if these p oints ar e p asse d in that or der in the pr o ess of moving along a ir le in a p ositive dir e tion. 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Figure 1. On the left a tree is D -planar. Theorem 2.1. If V ∗ ontains just two verti es, a tr e e T is D -planar. If ♯V ∗ ≥ 3 then a D -planarity of T is e quivalent to satisfying the fol lowing ondition: • for any e dge e ther e ar e exatly two p aths suh that they p ass thr ough an e dge e and onne t two adja ent verti es of V ∗ . 17 Pr o of. If ♯V ∗ = 2 , then T is homeomorphi to a segmen t and a set of its terminal v erties oinides with V ∗ = V ter , see Lemma 1.1. It is ob vious that there exists an em b edding ϕ : T → R 2 satisfying Denition 2.1 and a tree T is D -planar. Let ♯V ∗ ≥ 3 and T is D -planar. It means that there is an em b edding ϕ : T → R 2 whi h satises Denition 2.1. Let e ∈ E ( T ) b e an edge of T onneting v erties w 1 , w 2 ∈ V . W e x a p oin t x ∈ ϕ ( e ) \ { ϕ ( w 1 ) , ϕ ( w 2 ) } . A top ologial spae T is onedimensional ompat hene its homeomorphi image ϕ ( T ) is onedimensional [1℄. Then x ∈ ( R 2 \ ϕ ( T )) . It follo ws from (2) that x ∈ In t D 2 , therefore x ∈ ( R 2 \ ( ϕ ( T ) ∪ ∂ D 2 )) . By Lemma 1.1 there is a onneted omp onen t U j of a set R 2 \ ( ϕ ( T ) ∪ ∂ D 2 ) su h that a p oin t x b elongs to a b oundary of it. Corollary 1.1 states that ∂ U j ∩ ϕ ( T ) = ϕ ( P ( v j , v ′ j )) , where ϕ ( v j ) , ϕ ( v ′ j ) are adjaen t with resp et to a yli order of ϕ ( V ∗ ) indued from ∂ D 2 , see Remark 2.2. A ording to Denition 2.1, it is the same as v erties v j and v ′ j are adjaen t under a yli order C on V ∗ . So e ∈ P ( v j , v ′ j ) and v erties v j , v ′ j are adjaen t. It means that for an y edge of a D -planar tree T there is at least one path that satises the ondition of theorem. There exists an op en neigh b orho o d W = e \ { w 1 , w 2 } of a p oin t ϕ − 1 ( x ) in T that is homeomorphi to an in terv al. Using the ompatness of T \ W and theorem of Shenies [11, 13℄ w e an nd a neigh b orho o d U of x in R 2 \ ∂ D 2 and a homeomorphism h : U → In t D 2 su h that h ( x ) = ( 0 , 0) , h ◦ ϕ ( T ) = h ◦ ϕ ( W ) = ( − 1 , 1) × { 0 } . Let us designate U + = h − 1 ( { ( x, y ) ∈ In t D 2 | y > 0 } ) , U − = h − 1 ( { ( x, y ) ∈ In t D 2 | y < 0 } ) . It is lear that U ⊆ ϕ ( T ) ∪ U + ∪ U − . If for some omp onen t U k of R 2 \ ( ϕ ( T ) ∪ ∂ D 2 ) the in tersetions U + ∩ U k and U − ∩ U k are empt y , then x / ∈ U k and e / ∈ P ( v k , v ′ k ) in terms of Lemma 1.1. By the onstrution, the sets U + and U − are onneted and they b elong to R 2 \ ( ϕ ( T ) ∪ ∂ D 2 ) . Th us there are t w o omp onen ts U i and U j su h that U + ∈ U i , U − ∈ U j , x ∈ U i ∩ U j and e ∈ P ( v i , v ′ i ) ∩ P ( v j , v ′ j ) . By Corollaries 1.1 and 1.2 for an y edge of T there are no more then t w o paths su h that they onnet adjaen t v erties of V ∗ . In order to v erify that there are exatly t w o su h paths it is suien t to pro v e that U i 6 = U j . Supp ose that for some omp onen t U i of R 2 \ ( ϕ ( T ) ∪ ∂ D 2 ) w e get U \ ϕ ( T ) = U + ∪ U − ⊆ U i . An op en onneted subset U i of R 2 is path-onneted [2℄. 18 Denote a + 0 = (0 , 1 / 2) , a − 0 = (0 , − 1 / 2) ∈ In t D 2 , γ 0 = { 0 } × [ − 1 / 2 , 1 / 2] ⊆ In t D 2 , a + = h − 1 ( a + 0 ) , a − = h − 1 ( a − 0 ) ∈ U , γ = h − 1 ( γ 0 ) . It is ob vious that the p oin ts a + 0 and a − 0 are attainable from domains h ( U + ) \ γ 0 and h ( U − ) \ γ 0 b y a simple on tin uous urv e. Therefore the p oin ts a + and a − are attainable from the domain U i \ γ and there is a ut ˆ γ of U i \ γ b et w een a + and a − [11, 13℄. Then µ = γ ∪ ˆ γ is a simple lose urv e su h that µ ∩ ϕ ( T ) = { x } , µ \ { x } ⊆ U i and h ( µ ) ⊇ γ 0 . By Jordan's theorem µ b ounds an op en disk G [11, 13℄. The p oin t x do es not b elong to the ompat ˆ γ hene there exists its op en neigh b orho o d ˆ U ⊆ U su h that ˆ U ∩ ˆ γ = ∅ . Sine h maps a neigh b orho o d ˆ U of a p oin t x in to an op en neigh b orho o d of origin then there exists an ε ∈ (0 , 1 / 2) su h that a set Q 0 = { ( x, y ) ∈ D 2 | x 2 + y 2 < ε 2 } do es not in terset the set h ( ˆ γ ) . It follo ws that Q 0 ∩ h ( ϕ ( T ) ∪ ∂ D 2 ) = Q 0 ∩ h ◦ ϕ ( e ) = ( − ε, ε ) × { 0 } , Q 0 ∩ h ( µ ) = Q 0 ∩ γ 0 = { 0 } × ( − ε, ε ) . Denote Q = h − 1 ( Q 0 ) . Eviden tly , a set Q is an op en neigh b orho o d of x . Op en sets h − 1 ( { ( x, y ) ∈ Q 0 | x < 0 } ) and h − 1 ( { ( x, y ) ∈ Q 0 | x > 0 } ) are onneted and do not in terset the set µ . Therefore one of them m ust b e on tained in a disk G , another should b elong to an un b ounded domain R 2 \ G . Sets h − 1 (( − ε, 0) × { 0 } ) and h − 1 ((0 , ε ) × { 0 } ) b elong to the in tersetion of these domains with the image ϕ ( e ) of e . Hene ϕ ( e ) ∩ G 6 = ∅ and ϕ ( e ) ∩ R 2 \ G 6 = ∅ hold true. A segmen t ϕ ( e ) is divided b y x on t w o onneted ars that ha v e no ommon p oin ts with µ = ∂ G . Th us one of them should b elong to G and the other is on tained in R 2 \ G . Finally , the follo wing statemen t is true: either ϕ ( w 1 ) or ϕ ( w 2 ) b elongs to G and the other p oin t is on tained in R 2 \ G . Let ϕ ( w 1 ) ∈ G , ϕ ( w 2 ) ∈ R 2 \ G . By the onstrution, urv es ∂ D 2 and µ ha v e no ommon p oin ts sine either G ⊆ In t D 2 or In t D 2 ⊆ G . But ∅ 6 = ( γ ∩ Q ) ⊆ ( µ ∩ U i ) ⊆ ( µ ∩ Int D 2 ) . Therefore G ⊆ In t D 2 . Let us denote b y ˆ T a graph with a set of v erties V ( ˆ T ) = V ( T ) = V and a set of edges E ( ˆ T ) = E ( T ) \ { e } = E \ { e } . It is easy to sho w that the graph ˆ T has t w o onneted omp onen ts T 1 ∋ w 1 and T 2 ∋ w 2 . The images of them do not in terset with the urv e µ , therefore a set ϕ ( T 1 ) together with the p oin t ϕ ( w 1 ) b elongs to G ⊆ In t D 2 and ϕ ( T 2 ) ⊆ R 2 \ G . 19 By relation ϕ ( w 1 ) ∈ G ⊆ Int D 2 and Condition (2) , the v ertex w 1 has degree at least 2. Therefore it is adjaen t to at least one edge of T exept e that is an edge of T 1 . It means that a tree T 1 is non degenerated. Sine degrees of all other v erties of T 1 in T are the same as degrees in T 1 then V ter ( T 1 ) ⊆ V ter ( T ) ∪ { w 1 } . As w e kno w ♯V ter ( T 1 ) ≥ 2 whene there is w ∈ V ter ( T 1 ) ∩ V ter ( T ) . By the onstrution ϕ ( w ) ∈ G ⊆ I ntD 2 . On the other hand it follo ws from ( 1) and (2 ) that ϕ ( w ) ∈ ϕ ( V ∗ ) ⊆ ∂ D 2 . W e ha v e the on tradition with the assumption that U \ ϕ ( T ) ⊆ U i for some i . So, there are exatly t w o omp onen ts U i 6 = U j of a omplimen t R 2 \ ( ϕ ( T ) ∪ ∂ D 2 ) su h that the p oin t x ∈ ϕ ( e ) \ { ϕ ( w 1 ) , ϕ ( w 2 ) } whi h is on tained in the image of an edge e of T is a b oundary p oin t of. Consequen tly , b y Corollaries 1.1 and 1.2 there are exatly t w o paths su h that they pass through an arbitrary edge of T and onnet the adjaen t v erties of V ∗ . Let ♯V ∗ ≥ 3 and for an y e ∈ E ( T ) of T there are exatly t w o paths su h that they pass through this edge and onnet adjaen t v erties of V ∗ . W e should pro v e that the tree T is D -planar. A t rst w e onsider a relation C that is a full yli order on a set V ∗ . It generates a on v enien t relation ρ C on V ∗ , see Denition 1.5 . Let us examine a set of the direted paths P = { P ( v , v ′ ) | v ′ ρ C v } in T . By Denitions 1.1 and 1.5 t w o v erties v , v ′ ∈ V ∗ are adjaen t with resp et to a yli order C i either v ρ C v ′ or v ′ ρ C v is true. These orrelations an not hold true sim ultaneously , sine a pair of v erties v , v ′ w ould generate a ρ C -yle, see Denition 1.3, and this on tradits to Prop osition 1.3 and Corollary 1.4 sine ♯V ∗ ≥ 3 . It follo ws from the disussion ab o v e that for ev ery edge e of T there are exatly t w o paths of the set P passing through e . Let us onsider a binary relation ρ on the set V ∗ whi h is dened b y a orrelation (9) v ρv ′ ⇔ P ( v , v ′ ) ∈ P . Eviden tly , relation ρ is dual to the relation ρ C ( v 1 ρv 2 ⇔ v 2 ρ C v 1 ). Therefore b y Denition 1.2, ρ is the on v enien t relation on V ∗ . So, a minimal relation of equiv alene ˆ ρ on V ∗ on taining ρ oinides with a minimal relation of equiv alene ˆ ρ C on V ∗ on taining ρ C . Th us the elemen ts of the set V ∗ generate a ρ -yle, see Prop osition 1.3 and Corollary 1.3. Let e b e an edge of the tree T . W e should pro v e that those t w o direted paths of the set P that on tain e pass through e in opp osite diretions. 20 Let us onsider a binary relation µ e on V ∗ that is dened as follo ws v µ e v ′ ⇔ P ( v , v ′ ) ∈ P i e / ∈ P ( v , v ′ ) . It is easy to see that a diagram of the relation µ e an b e obtained from a diagram of ρ b y remo ving t w o pairs of v erties of V ∗ orresp onding to paths of P whi h pass through e . Let ( v 1 , v ′ 1 ) and ( v 2 , v ′ 2 ) b e su h pairs. Therefore the relation µ e satises the onditions of Lemma 1.4 . By this Lemma a minimal relation of equiv alene ˆ µ e on taining µ e has t w o lasses of equiv alene B 1 , B 2 and v 1 ∈ B 1 , v 2 ∈ B 2 . Let w , w ′ ∈ V b e the ends of e . Let us onsider a subgraph T ′ of the tree T su h that V ( T ′ ) = V ( T ) and E ( T ′ ) = E ( T ) \ { e } . It is lear that the v erties w and w ′ b elong to dieren t onneted omp onen ts of a graph T ′ (if there exists a path P in T ′ su h that it onnets them then these v erties an b e onneted b y t w o dieren t paths P and P ′ = { e } in the tree T ). W e denote these omp onen ts b y T w and T w ′ . Supp ose that for v erties v , v ′ ∈ V there is an direted path P ( v , v ′ ) passing through e . Let it rst passes through the v ertex w and then though w ′ . Then paths P ( v , w ) and P ( w ′ , v ′ ) b elong to T ′ , so v ∈ T w , v ′ ∈ T w ′ . In ase when the path P ( v , v ′ ) rst passes through w ′ and then through w w e ha v e v ′ ∈ T w and v ∈ T w ′ . It is easy to see that ev ery lass of equiv alene of the relation ˆ µ e b elongs to the unique onneted omp onen t of the set T ′ . By the onstrution dieren t lasses of equiv alene ha v e to b elong to the dieren t omp onen ts of T ′ . So, w e onlude that either B 1 ⊆ T w and B 2 ⊆ T w ′ or B 1 ⊆ T w ′ and B 2 ⊆ T w . Supp ose that rst pair of inequalities holds true. If the direted paths P ( v 1 , v ′ 1 ) and P ( v 2 , v ′ 2 ) pass through e in the same diretion, then P ( v 1 , w ) ∪ P ( v 2 , w ) ⊆ T w and v 2 ∈ T w . By the onstrution T w ∩ V ∗ = B 1 th us v 2 ∈ B 1 . But it is a on tradition to Lemma 1.4 . So, the paths P ( v 1 , v ′ 1 ) and P ( v 2 , v ′ 2 ) pass through e in the opp osite diretions. The ase B 1 ⊆ T w ′ , B 2 ⊆ T w is onsidered similarly . Let us onstrut an em b edding of T in to orien ted disk D 2 . Let D 2 b e an orien ted disk (losed disk with a xed orien tation on the b ound- ary), I = [0 , 1] an direted segmen t and ψ : I → D 2 an em b edding su h that ψ ( I ) ⊆ ∂ D 2 . The diretion of a segmen t is said to b e o or dinate d with the orien- tation of disk if a diretion of passing along the simple on tin uous urv e ψ ( I ) from the origin ψ (0) to the end ψ (1) oinides with giv en orien tation of the b oundary ∂ D 2 . Ev ery direted path in T is top ologially a losed segmen t th us for direted path P ( v , v ′ ) with the origin v and the end v ′ there exists an em b edding Φ P ( v ,v ′ ) : P ( v , v ′ ) → D 2 su h that Φ P ( v ,v ′ ) ( P ( v , v ′ )) ⊆ ∂ D 2 and a diretion of P ( v , v ′ ) is o ordinated with the orien tation of D 2 . 21 W e x a disjoin t union of losed orien ted disks F P ∈P D P and a set of the em- b eddings Φ P ( v ,v ′ ) : P ( v , v ′ ) → D P ( v ,v ′ ) , (10) Φ P ( v ,v ′ ) ( P ( v , v ′ )) ⊆ ∂ D P ( v ,v ′ ) , P ( v , v ′ ) ∈ P , su h that the diretions of paths P ( v , v ′ ) ∈ P are o ordinated with the orien tations of orresp onding disks. Let us onsider a spae ˜ D = T ⊔ G P ∈P D P . All maps Φ P , P ∈ P are injetiv e therefore a family of sets F x = { x } S P ∈P : x ∈ P Φ P ( x ) , x ∈ T , { x } , x ∈ S P ∈P D P \ Φ P ( P ) . generates a partition f of the spae ˜ D . W e onsider a fator-spae D of ˜ D o v er partition f and a pro jetion map pr : ˜ D → D . Let us pro v e that D is homeomorphi to a disk, the orien tations of disks D P , P ∈ P giv e some orien tation on D and a map ϕ = pr | T : T → D onforms to the onditions of Denition 2.1. A t rst w e in v estigate some prop erties of the spae D and the pro jetion pr . 2.1. The mapping pr is losed. Reall that a set is alled satur ate d o v er partition f if it onsists of en tire elemen ts of that partition. T op ology of spae D is a fator-top ology (a set A is losed in D i its full preimage pr − 1 ( A ) is losed in ˜ D ). F or pro of of losure of a pro jetion map pr it is suien t to he k that for an y losed subset K of the spae ˜ D minimal saturated set ˜ K = pr − 1 (pr( K )) on taining K is also losed. F rom the denition of partition f it follo ws that K = ( K ∩ T ) ⊔ G P ∈P ( K ∩ D P ) , ˜ K = ( K ∩ T ) ⊔ G P ∈P (( K ∩ D P ) ∪ Φ P ( K ∩ P )) . (11) Sets T , P , D P , P ∈ P are ompats and all maps Φ P are homeomorphisms on to their images. Th us all sets K ∩ T , K ∩ D P , Φ P ( K ∩ P ) , P ∈ P , are ompats. The graph T is nite hene ♯ P < ∞ and the union on the righ t of (11 ) is nite. The set ˜ K is a ompat, so it is losed. W e remark that w e iniden tally v eried that the spae ˜ D is ompat. 22 2.2. The spae D is a ompatum. D is the ompat spae as a fator-spae of ompat spae ˜ D . Compatum ˜ D is the normal top ologial spae and a fator-spae of a normal spae o v er losed partition is a normal spae, see [4℄. Th us D is a normal spae, in partiularly , D is Hausdor spae. Therefore D is ompatum. 2.3. Map ϕ = pr | T : T → D is the em b edding. By denition, F x ∩ T = { x } for ev ery x ∈ T , hene ϕ is an injetiv e map. The spae T is ompat and the spae D is Hausdor th us ϕ is homeomorphism on to its image, see [4℄. 2.4. F or ev ery P ∈ P a map pr D P : D P → D is an em b edding. By denition, for x ∈ D P w e get (12) D P ∩ F x = ( Φ P (Φ − 1 P ( x )) , x ∈ Φ P ( P ) , { x } , x ∈ D P \ Φ P ( P ) . The map Φ P is injetiv e hene Φ P (Φ − 1 P ( x )) = { x } , x ∈ Φ P ( P ) . Finally , F x ∩ D P = { x } for ev ery x ∈ D P and a on tin uous map pr D P is injetiv e. Th us it is a homeomorphism of ompat D P on to its image. 2.5. F or ev ery P ∈ P a set pr( D P \ Φ P ( P )) is op en in D and has no ommon p oin ts with a set pr( ˜ D \ ( D P \ Φ P ( P ))) . Let P ∈ P . The set D P is op en-losed in the spae ˜ D , hene an op en set D P \ Φ P ( P ) in D P is also op en in ˜ D . This set is saturated b y denition. Therefore D P \ Φ P ( P ) = pr − 1 (pr( D P \ Φ P ( P ))) and a set pr( D P \ Φ P ( P )) is op en in the fator-spae D . It follo ws from the disussion ab o v e that a set ˜ D \ ( D P \ Φ P ( P )) is also saturated and it has no ommon p oin ts with D P \ Φ P ( P ) . Th us pr( D P \ Φ P ( P )) ∩ pr( ˜ D \ ( D P \ Φ P ( P ))) = ∅ . 2.6. Let e ∈ E b e an y edge of the tree T , p oin ts w 1 , w 2 b e the ends of e and P ′ , P ′′ ∈ P b e paths in P that pass through e . W e designate e 0 = e \ { w 1 , w 2 } , D 0 P ′ = D P ′ \ ∂ D P ′ ⊆ [ p ∈P D P \ Φ P ( P ) , D 0 P ′′ = D P ′′ \ ∂ D P ′′ ⊆ [ p ∈P D P \ Φ P ( P ) , ˜ U = ( D 0 P ′ ∪ Φ P ′ ( e 0 )) ⊔ ( D 0 P ′′ ∪ Φ P ′′ ( e 0 )) ⊔ e 0 , U = pr( ˜ U ) . U is the op en neigh b orho o d of a set pr( e 0 ) in the spae D , it is homeomorphi to op en disk and is divided b y a set pr( e 0 ) on to t w o onneted omp onen ts pr( D 0 P ′ ) and pr( D 0 P ′′ ) . 23 T o pro v e this w e should remark that sets e 0 , D 0 P ′ and D 0 P ′′ are op en in ˜ D . By denition of partition f for ev ery x ∈ e 0 w e get F x = { x, Φ P ′ ( x ) , Φ P ′′ ( x ) } sine the set ˜ U is saturated. The set ˜ U is op en in ˜ D . Really , in the rst plae e 0 is an op en subset of T , seondly , Φ P ′ ( e 0 ) is an op en subset of losed subspae Φ P ′ ( P ′ ) of spae D P ′ , therefore, Φ P ′ ( P ′ ) \ Φ P ′ ( e 0 ) is a losed subset D P ′ . Let us remark that ars Φ P ′ ( P ′ ) and ∂ D P ′ \ Φ P ′ ( P ′ ) ha v e Φ P ′ - images of endp oin ts of the path P ′ as ommon ends, th us ∂ D P ′ \ Φ P ′ ( P ′ ) ∩ Φ P ′ ( e 0 ) = ∅ and follo wing onditions hold true ∂ D P ′ \ Φ P ′ ( e 0 ) = ( ∂ D P ′ \ Φ P ′ ( P ′ )) ∪ (Φ P ′ ( P ′ )) \ Φ P ′ ( e 0 )) = = ( ∂ D P ′ \ Φ P ′ ( P ′ )) ∪ (Φ P ′ ( P ′ )) \ Φ P ′ ( e 0 )) . So, a set ∂ D P ′ \ Φ P ′ ( e 0 ) is losed in D P ′ and a set D 0 P ′ ∪ Φ P ′ ( e 0 ) = D P ′ \ ( ∂ D P ′ \ Φ P ′ ( e 0 )) is op en in D P ′ . Similarly , a set D 0 P ′′ ∪ Φ P ′′ ( e 0 ) is op en in D P ′′ . Sets T , D P ′ and D P ′′ are op en-losed in spae ˜ D . Th us the set ˜ U is op en in ˜ D . Finally , the set U = pr( ˜ U ) is op en in D . This set is a result of gluing U ∼ = ( D 0 P ′′ ∪ Φ P ′′ ( e 0 )) ∪ α ( D 0 P ′ ∪ Φ P ′ ( e 0 )) , α = Φ P ′′ ◦ Φ − 1 P ′ : Φ P ′ ( e 0 ) → Φ P ′′ ( e 0 ) . A map α is a omp osition of homeomorphisms. Therefore U is homeomorphi to op en disk and is divided b y pr( e 0 ) on to t w o onneted omp onen ts pr( D 0 P ′ ) and pr( D 0 P ′′ ) . 2.7. F or an y P 1 , . . . , P n ∈ P a b oundary F r D n of a set D n = pr( S n i =1 D P i ) in the spae D b elongs to pr( S n i =1 P i ) = pr( T ) ∩ D n . It follo ws from prop ert y 2.5 that F r pr( D P i ) ⊆ pr(Φ P i ( P i )) = pr( P i ) for an y i ∈ { 1 , . . . , n } . Hene F r D n ⊆ n [ i =1 F r pr( D P i ) ⊆ n [ i =1 pr( P i ) = pr n [ i =1 P i . 2.8. Let P 1 , . . . , P n ∈ P , ˜ D n = S n i =1 D P i , D n = pr( ˜ D n ) . Let e b e an edge of T su h that pr( e 0 ) ∩ D n 6 = ∅ , where e 0 is an edge e without ends. A set pr( e 0 ) b elongs to In t D n i at least one p oin t y ∈ pr( e 0 ) has a neigh b orho o d in D n whi h is homeomorphi to op en disk. Otherwise, a set pr( e 0 ) b elongs to F r D n . If pr( e 0 ) ⊆ Int D n then a set pr( e 0 ) has a neigh b orho o d in D n whi h is homeo- morphi to op en disk and b oth paths P ′ , P ′′ ∈ P passing through e b elong to a set { P 1 , . . . , P n } . If pr( e 0 ) ⊆ F r D n , then exatly one of them b elongs to { P 1 , . . . , P n } . Supp ose that paths P ′ , P ′′ ∈ P pass through the edge e . By the denition pr( e 0 ) ⊆ D n ∩ pr( T ) = pr( S n i =1 P i ) , so at least one of them b elongs to { P 1 , . . . , P n } . W e onsider t w o p ossibilities. 24 W e assume that P ′ = P k , P ′′ = P s , k , s ∈ { 1 , . . . , n } . Then a set U = pr( ( D 0 P ′ ∪ Φ P ′ ( e 0 )) ∪ ( D 0 P ′′ ∪ Φ P ′′ ( e 0 )) ∪ e 0 ) ⊆ D n is an op en neigh b orho o d of pr( e 0 ) that is homeomorphi to an op en disk, see 2.6. Let P ′ ∈ { P 1 , . . . , P n } , P ′′ / ∈ { P 1 , . . . , P n } . In this ase U = U ′ ∪ U ′′ ∪ e 0 , U ′ = pr( D 0 P ′ ) ⊆ D n but a set U ′′ = pr( D 0 P ′′ ) has no ommon p oin ts with D n , therefore pr( e 0 ) ⊆ F r D n . Supp ose that for some y ∈ pr( e 0 ) in D n there exists an neigh b orho o d W y ∈ D n homeomorphi to op en t w o dimensional disk. By using theorem of Shenies [11, 13℄ w e an nd a small neigh b orho o d ˆ W y of y in D n su h that it is homeomorphi to an op en disk and satises follo wing onditions: • A set ˆ W y in the spae D n is homeomorphi to a losed disk and is separated from ompats pr( T \ e 0 ) and D \ U . • ˆ W y in tersets pr( e 0 ) b y a onneted segmen t that is a ut of the disk ˆ W y . Then the set pr( e 0 ) divides ˆ W y on to t w o onneted omp onen ts W 1 ∪ W 2 = ˆ W y \ pr( e 0 ) , W 1 ∩ W 2 = ∅ su h that W 1 ∩ W 2 ∋ y . By the onstrution ˆ W y ⊆ U ∩ D n and W 1 , W 2 ⊆ ( U ∩ D n ) \ pr( T ) . Let us remind that P ′ ∈ { P 1 , . . . , P n } , th us pr( D P ′ ) = D ′ ⊆ D n . Similarly , P ′′ / ∈ { P 1 , . . . , P n } hene pr( D 0 P ′′ ) ∩ D n = ∅ , see prop ert y 2.5 . Therefore U ∩ D n = U ′ ∪ pr( e 0 ) , where U ′ = pr( D 0 P ′ ) , see prop ert y 2.6 , and U ∩ D n ∩ pr( T ) = pr( e 0 ) ⊆ ∂ D ′ , where D ′ = pr( D P ′ ) . Th us y ∈ ∂ D ′ and the set ˆ W y is the op en neigh b orho o d of y in losed disk D ′ and ˆ W y ∩ ( D ′ \ ∂ D ′ ) = W 1 ∪ W 2 , y ∈ W 1 ∩ W 2 . By Lemma 1.5 w e an onlude that W 1 = W 2 but it on tradits to the assumption that W 1 ∩ W 2 = ∅ . So, if { P ′ , P ′′ } * { P 1 , . . . , P n } , then there is no y ∈ pr( e 0 ) that has an op en neigh b orho o d in D n , whi h is homeomorphi to op en disk. 2.9. Let P 1 , . . . , P n ∈ P . Let us desrib e a struture of b oundary F r D n of D n = pr( S n i =1 D P i ) in D . Denote b y E n ⊆ E a set of all edges of the tree T su h that exatly one of t w o paths P ′ , P ′′ ∈ P passing through e ∈ E n b elongs to { P 1 , . . . , P n } . As w e kno w, see Condition 2.8, pr( E n ) ⊆ F r D n and if for some edge e ∈ E w e get e / ∈ E n , then F r D n ∩ pr( e ) ⊆ { v ′ , v ′′ } , where v ′ , v ′′ ∈ V are ends of e . Similarly , denote b y V n ⊆ V a set of all v erties of T su h that for a v ertex v ∈ V n the follo wing ondition satises: pr( v ) ∈ D n and all edges that are adjaen t to v b elong to E \ E n . It is easy to sho w that the set V n is disreet and pr( E n ) ∩ pr ( V n ) = ∅ . F rom the disussion ab o v e and Condition 2.7 it follo ws that (13) pr( E n ) ⊆ F r ( D n ) ⊆ (pr( E n ) ∪ pr( V n )) . 25 2.10. Let P 1 , . . . , P n ∈ P . A set D n = pr( S n i =1 D P i ) is onneted i then S n i =1 P i is a onneted subgraph of the tree T . Let S n i =1 P i = T ′ is a onneted subgraph of T . Then D n = pr( T ′ ) ∪ S n i =1 pr( D P i ) , all sets pr( T ′ ) , pr( D P i ) , i ∈ { 1 , . . . , n } are onneted and pr( T ′ ) ∩ pr( D P i ) 6 = ∅ , i ∈ { 1 , . . . , n } . Hene the set D n is onneted. Next, let S n i =1 D P i = T ′ ∪ T ′′ , T ′ ∩ T ′′ = ∅ and sets T ′ , T ′′ are nonempt y and losed. Ev ery set P i , i ∈ { 1 , . . . , n } is onneted, therefore, either P i ∈ T ′ or P i ∈ T ′′ . Without loss of generalit y w e an hange indexing of the elemen ts of { P 1 , . . . , P n } in su h w a y that for some s ∈ { 1 , . . . , n − 1 } the follo wing onditions are satised T ′ = s [ i =1 P i , T ′′ = n [ i = s +1 P i . Ev ery set ˜ D ′ = T ′ ∪ s [ i =1 D P i , ˜ D ′′ = T ′′ n [ i = s +1 D P i , is losed, whene sets D ′ = pr ( ˜ D ′ ) i D ′′ = pr( ˜ D ′′ ) are losed, see Condition 2.1 . By the onstrution ˜ D ′ ∩ ˜ D ′′ = ∅ . Let y ∈ D ′ ∩ D ′′ . A map pr is injetiv e b y denition on the set pr − 1 ( D \ pr( T )) and sets ˜ D ′ and ˜ D ′′ do not in terset on pr − 1 ( D \ pr( T )) , th us y ∈ pr( T ) . Hene y ∈ pr( T ∩ ˜ D ′ ) ∩ pr( T ∩ ˜ D ′′ ) = pr ( T ′ ) ∩ pr( T ′′ ) . But as w e kno w, see Condition 2.3, the map ϕ = pr | T is bijetiv e, therefore pr( T ′ ) ∩ pr( T ′′ ) = pr( T ′ ∩ T ′′ ) = ∅ . W e get a on tradition, th us D ′ ∩ D ′′ = ∅ . Hene D n = D ′ ⊔ D ′′ an sets D ′ , D ′′ are losed and nonempt y . Therefore the set D n is not onneted. Finally let us pro v e a D -planarit y of the tree T . Let for some n , 1 ≤ n < ♯ P direted paths P 1 = P ( v 1 , v ′ 1 ) , . . . , P n = P ( v n , v ′ n ) ∈ P are xed and ˜ D n = S n i =1 D P i , D n = pr( ˜ D n ) . F or ev ery i ∈ { 1 , . . . , n } w e denote b y ˜ γ i an direted ar of ∂ D P i from p oin t Φ P i ( v ′ i ) to Φ P i ( v i ) whi h has no other ommon p oin ts with an ar Φ P i ( P i ) . Supp ose that the ob jets under onsideration omply with follo wing onditions. (i) A spae D n is homeomorphi to a lose t w o-dimensional disk. (ii) There exists at least one edge e ∈ S n i =1 P i su h that its image pr( e ) is on tained in a b oundary irle ∂ D n of D n . (iii) A disk D n is orien ted in the follo wing w a y: for ev ery i ∈ { 1 , . . . , n } and ev ery edge e ∈ P i su h that pr( e ) b elongs to ∂ D n an orien tation of e generated b y the diretion of P i = P ( v i , v ′ i ) maps b y pr on to an orien tation of D n . (iv) F or ev ery i ∈ { 1 , . . . , n } an ar γ i = pr( ˜ γ i ) onnets a p oin t pr( v ′ i ) with a p oin t pr( v i ) and has no other ommon p oin ts with a set pr( T ) and orien- tation of this ar is onsisten t with the orien tation of D n . 26 W e should remark that for n = 1 and an y path P = P 1 ∈ P if w e tak e an orien tation on D 1 = pr( D p ) indued from D P b y using pr , then Conditions (i) (iv) alw a ys hold true. By the onstrution, Conditions (iii) and (iv) are true , (i) follo ws from Condition 2.4 , (ii) follo ws from Condition 2.8. W e also remark that it follo ws from Condition 2.8 that an edge e ∈ S n i =1 P i b elongs to ∂ D n of D n i e ∈ E n . Th us Condition (iii) is w ell-p osed. As w ell all b oundary p oin ts of D n in the spae D p ossibly exept a nite n um b er of isolated p oin ts from the set pr( V n ) b elong to ∂ D n . Let an edge e ∈ S n i =1 P i satises Condition (ii). Then e ∈ E n and there is the unique path P n +1 = P ( v n +1 , v ′ n +1 ) ∈ P \ { P 1 , . . . , P n } su h that it passes through the edge e . Let e ∈ P l , where P l ∈ { P 1 , . . . , P n } is the seond path among t w o paths from the set P whi h passes through the edge e . Let us onsider a disk D P n +1 and its image D ′ = pr( D P n +1 ) . By Condition 2.4 it is also the losed disk. Let Γ = D n ∩ D ′ . It is ob vious that Γ is losed. By Condition 2.5 a set pr( D P n +1 \ Φ P n +1 ( P n +1 )) is op en in D and do es not in terset D n . It follo ws from Condition 2.4 that pr( D P n +1 \ Φ P n +1 ( P n +1 )) = pr( D P n +1 ) \ pr ◦ Φ P n +1 ( P n +1 ) = D ′ \ pr( P n +1 ) , pr(Φ P n +1 ( P n +1 )) = pr( P n +1 ) ⊆ ( D ′ \ pr( P n +1 )) . Therefore Γ = F r D n ∩ F r D ′ ⊆ pr( P n +1 ) . Let us apply Condition 2.9 to D n and D ′ . By (13 ) the set Γ onsists of images of edges whi h b elong to the path P n +1 and p ossibly from a n um b er of images of v erties of a tree T . Let us he k that the set Γ is onneted. If it is not the ase it follo ws from what w e said ab o v e that there are t w o v erties w 1 , w 2 ∈ V , w 1 6 = w 2 of T su h that they b elong to the path P n +1 and a pro jetion of a path P ( w 1 , w 2 ) ⊆ P n +1 whi h onnets them in T in tersets Γ b y a set { pr( w 1 ) , pr( w 2 ) } . Then pr( P ( w 1 , w 2 )) ∩ D n = { w 1 , w 2 } . On the other hand, the set D n is onneted th us T ′ = S n i =1 P i is a onneted subgraph of T , see Condition 2.10 . F rom Condition 2.7 it follo ws that w 1 , w 2 ∈ V ( T ′ ) , therefore there is a path P ′ ( w 1 , w 2 ) onneting them in T ′ . This path has to onnet w 1 with w 2 in T . But pr( P ′ ( w 1 , w 2 )) ⊆ D n hene P ′ ( w 1 , w 2 ) 6 = P ( w 1 , w 2 ) . So, v erties w 1 and w 2 of T an b e onneted in T b y t w o dieren t paths whi h is imp ossible in the tree T . This on tradition pro v es that Γ is onneted. It follo ws from the onnetedness of Γ and from the inlusion pr( e ) ⊆ Γ ∩ pr ( E n ) that Γ ⊆ pr( E n ) . Th us Γ ⊆ ∂ D n ∩ ∂ D ′ , where ∂ D ′ = pr( ∂ D P n +1 ) is a b oundary irle of the disk D ′ . 27 By the disussion ab o v e and from Γ ⊆ pr( P n +1 ) it is easy to understand that Γ = pr( P ( v , v ′ )) for some v , v ′ ∈ V ∩ P n +1 , v 6 = v ′ . It is ob vious that P ( v , v ′ ) is homeomorphi to a losed segmen t. F rom the Conditions 2.3 and 2.4 it follo ws that it is em b edded in to a b oundary irles ∂ D n and ∂ D ′ b y means of maps ψ n = pr P ( v ,v ′ ) : P ( v , v ′ ) → D n , ψ ′ = pr ◦ Φ P n +1 : P ( v , v ′ ) → D ′ . Therefore, a set D n +1 = D n ∪ D ′ ∼ = D n ∪ ψ D ′ , ψ = ψ n ◦ ( ψ ′ ) − 1 , is a result of a gluing of losed disks D n and D ′ b y a segmen t that is em b edded in to the b oundary irles of these disks. Consequen tly the set D n +1 is homeomorphi to a losed disk. Let us denote ˜ D n +1 = S n +1 i =1 D P i . It is lear that D n +1 = pr n [ i =1 D P i ∪ pr( D P n +1 ) = pr n +1 [ i =1 D P i = pr( ˜ D n +1 ) . Hene the spae D n +1 onstruted aording to the set { P 1 , . . . , P n +1 } satises Condition (i). Disks D n and D ′ are orien ted. The orien tation of D ′ is generated b y an orien- tation of D P n +1 b y means of the map pr . By Condition (iii) applied to D n and D ′ w e get t w o orien tations on e . One of them is indued from an orien tation of P l ⊇ e and is o ordinated with orien tation of D n . Another is generated b y diretion of P n +1 and is onsisten t with an orien tation of D ′ . As w e said ab o v e the direted paths P l , P n +1 ∈ P on taining an edge e ha v e to pass through e in the opp osite diretions. Therefore the orien tations indued on Γ from D n and D ′ are opp osite. Hene the orien tations of D n and D ′ are o ordinated and generate an orien tation of D n +1 . It omplies with the follo wing ondition • for an y simple ar α : I → ∂ D n ∩ ∂ D n +1 an orien tation of α is onsisten t with orien tation of D n +1 i an orien tation α is o ordinated with orien tation of D n ; • for an y simple ar β : I → ∂ D ′ ∩ ∂ D n +1 an orien tation of β is onsisten t with an orien tation of D n +1 i it is o ordinated with an orien tation a disk D ′ . Disks D n and D ′ satisfy Conditions (iii) and (iv). So, aording to what has b eing said D n +1 also satises Conditions (iii) and (iv). 28 Supp ose that the set D n +1 do es not satisfy Condition (ii). Then E n +1 = ∅ , see Condition 2.9 and Remark (iii), and ∂ D n +1 ∩ pr( T ) ⊆ pr( V ) . Th us a set ∂ D n +1 ∩ pr( T ) is nite. The follo wing orrelations are impliated from Condition 2.5 ∂ D n +1 \ pr( T ) ⊆ pr n +1 [ i =1 ( D P i \ Φ P i ( P i )) = n +1 [ i =1 pr( D P i \ Φ P i ( P i )) . F rom Condition 2.4 it follo ws that for ev ery i ∈ { 1 , . . . , n +1 } a set pr( D P i \ ∂ D P i ) ⊆ D n +1 is homeomorphi to an op en disk. Hene n +1 [ i =1 pr( D P i \ ∂ D P i ) ⊆ D n +1 \ ∂ D n +1 . F rom this orrelation it follo ws, see Condition (iv), that ∂ D n +1 \ pr( T ) ⊆ " n +1 [ i =1 pr( D P i \ ∂ D P i ) ∪ pr( ∂ D P i \ Φ P i ( P i )) # ∩ ∂ D n +1 = = n +1 [ i =1 pr( ∂ D P i \ Φ P i ( P i )) ⊆ n +1 [ i =1 pr( ˜ γ i ) = n +1 [ i =1 γ i . A set S n +1 i =1 γ i is losed in D hene it is also losed in ∂ D n +1 . Therefore, a set ∂ D n +1 \ S n +1 i =1 γ i ha v e to b e an op en subset of a spae ∂ D n +1 . But ∂ D n +1 \ n +1 [ i =1 γ i ⊆ ∂ D n +1 ∩ pr( T ) ⊆ pr( V ) and this set is nite. Consequen tly , ∂ D n +1 = n +1 [ i =1 γ i . F rom Condition (iv) it easily follo ws that op en ars γ i \ { pr( v i ) , pr( v ′ i ) } , i ∈ { 1 , . . . , n + 1 } are pairwise disjoin t. Therefore ev ery p oin t of a set ∂ D n +1 ∩ pr ( T ) = S n +1 i =1 { pr( v i ) , pr( v ′ i ) } is a ommon b oundary p oin t of exatly t w o ars of the family { γ i } n +1 i =1 . It follo ws from the hoie of an orien tation of ars γ i , i ∈ { 1 , . . . , n + 1 } that if for some s , r ∈ { 1 , . . . , n + 1 } either v s = v r or v ′ s = v ′ r is true, then s = r . Th us for ev ery i ∈ { 1 , . . . , n + 1 } there is the unique j ( i ) ∈ { 1 , . . . , n + 1 } , su h that v i = v ′ j and if r 6 = s then j ( r ) 6 = j ( s ) . W e also remark that b y the onstrution n ≥ 1 , th us n + 1 ≥ 2 and j ( i ) 6 = i , i ∈ { 1 , . . . , n + 1 } . Therefore, on the set { 1 , . . . , n + 1 } there is a transp osition σ without x p oin ts su h that v i = v ′ σ ( i ) , i ∈ { 1 , . . . , n + 1 } . Let σ = c 1 · · · c k b e a deomp osition of σ in to indep enden t yles. Let c 1 = ( i 1 . . . i m ) . Then v i 1 = v ′ i 2 , . . . v i m − 1 = v ′ i m , v i m = v ′ i 1 . 29 F rom the denition of the set P w e get v ′ i ρ C v i , i ∈ { 1 , . . . , n + 1 } , sine P i = P ( v i , v ′ i ) ∈ P . So, it is true that v i 1 ρ C v i 2 , . . . , v i m − 1 ρ C v i m , v i m ρ C v i 1 , th us v erties of the set M 1 = { v i 1 , . . . , v i m } generate a ρ C -yle, see Denition 1.3. F rom Corollary 1.4 it is follo ws that the set M 1 is a lass of equiv alene of a minimal equiv alene relation ˆ ρ C whi h on tains the relation ρ C . By Prop osition 1.3 and Corollary 1.4 the relation ˆ ρ C has the unique lass of equiv alene V ∗ . Hene M 1 = V ∗ , σ = c 1 , n + 1 = ♯V ∗ = ♯ P and D n +1 = D . F rom what w as said ab o v e it follo ws that for n + 1 < ♯ P the disk D n +1 satises Condition (ii). Th us, for n + 1 < ♯ P the disk D n +1 satises (i)(iv), but for n + 1 = ♯ P it omplies with onditions (i) and (iv). Finally , starting from an y path P = P 1 = P ( v 1 , v ′ 1 ) ∈ P , w e an sort out elemen ts of a set P = { P 1 = P ( v 1 , v ′ 1 ) , . . . , P N = P ( v N , v ′ N ) } in a nite n um b er of steps so that for ev ery set D n = pr n [ i =1 D P i , n ∈ { 1 , . . . , N − 1 } , the onditions (i)(iv) are true and for the set D N = pr N [ i =1 D P i = pr [ P ∈P D P = pr( ˜ D ) = D onditions (i) i (iv) are also true. Th us D N = D is losed orien ted t w o-dimensional disk, ϕ = pr | T : T → D is an em b edding, see Condition 2.3 . F or ev ery edge e ∈ E b oth paths of P passing through this edge b elong to a set { P 1 , . . . , P N } , th us E N = ∅ and ∂ D = S N i =1 γ i with op en ars γ i \ { pr( v i ) , pr( v ′ i ) } are pairwise disjoin t. It is lear that ϕ ( T ) ∩ ∂ D = N [ i =1 { pr( v i ) , pr( v ′ i ) } = [ P ( v ,v ′ ) ∈P { pr( v ) , pr( v ′ ) } = V ∗ . An orien tation of D generates some yli order O on the set pr( V ∗ ) . A map ϕ 0 = ϕ | V ∗ : V ∗ → pr( V ∗ ) is bijetiv e, therefore, a map ϕ − 1 0 generates on the set V ∗ some yli order C ′ whi h is an isomorphi image of a yli order O ( C ′ ( v 1 , v 2 , v 3 ) ⇔ O (pr( v 1 ) , pr( v 2 ) , pr( v 3 )) ). W e indue a on v enien t relation ρ C ′ on V ∗ , see Denition 1.5 . F rom Con- dition (iv) it follo ws that for ev ery i ∈ { 1 , . . . , N } w e ha v e v ′ i ρ C ′ v i . On the other hand, b y denition of the set P it follo ws that v ′ ρ C v i P ( v , v ′ ) ∈ P . But 30 P = { P 1 , . . . , P N } , hene if P ( v , v ′ ) ∈ P , then P ( v , v ′ ) = P i = P ( v i , v ′ i ) for some i ∈ { 1 , . . . , N } . Therefore the follo wing onditions hold true v ′ ρ C v ⇒ v ′ ρ C ′ v , v , v ′ ∈ V ∗ , and the relation ρ C ′ on tains ρ C . With the help of on v enien t relations ρ C and ρ C ′ w e an indue on V ∗ the relations of yli orders C ρ C and C ρ C ′ , resp etiv ely , see Denition 1.6 and Prop o- sition 1.4. F rom Denition 1.6 it is easily follo ws that if ρ C ′ on tains ρ C then C ρ C ′ on tains C ρ C . In other w ords, an iden tial map I d V ∗ is monomorphism of yli order C ρ C on to C ρ C ′ , see Denition 1.7 . F rom Lemma 1.3 it follo ws that C ρ C = C and C ρ C ′ = C ′ , hene the map I d V ∗ is monomorphism of the yli order C on to C ′ . Lemma 1.2 implies that the map I d V ∗ is an isomorphism of yli order C on to C ′ . By the onstrution a map ϕ − 1 0 is an isomorphism of yli order O on to C ′ th us ϕ 0 is an isomorphism of yli order C = C ′ on to a yli order O whi h is indued on to ϕ ( V ∗ ) from an orien ted irle ∂ D . Finally , the map ϕ satises all onditions of Denition 2.1 and a tree T is D -planar. Referenes [1℄ Kur atovskiy K. T op ology . V ol. I. New edition, revised and augmen ted. 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