An analogue of a theorem of Steinitz for ball polyhedra in R 3
AAN ANALOGUE OF A THEOREM OF STEINITZ FOR BALLPOLYHEDRA IN R SAMI MEZAL ALMOHAMMAD , ZSOLT L ´ANGI, AND M ´ARTON NASZ ´ODI
Abstract.
Steinitz’s theorem states that a graph G is the edge-graph of a 3-dimensionalconvex polyhedron if and only if, G is simple, plane and 3-connected. We prove an ana-logue of this theorem for ball polyhedra, that is, for intersections of finitely many unitballs in R . Introduction
Our work takes place in Euclidean 3-space. For the closed ball of radius ρ centeredat x ∈ R , we use the notation B [ x, ρ ] := { y ∈ R : d ( x, y ) ≤ ρ } . The 2-dimensionalsphere (the boundary of a closed ball) is denoted by S ( x, ρ ) := { y ∈ R : d ( x, y ) = ρ } .For brevity, we set B [ x ] := B [ x, S ( x ) := S ( x,
1) and for a set X ⊆ R , we write B [ X ] := (cid:84) x ∈ X B [ x ].Let X ⊂ R be a finite, nonempty set contained in a ball of radius less than 1. Theset P = B [ X ] is called a ball polyhedron . For any x ∈ X , we call B [ x ] a generating ball of P and S ( x ) a generating sphere of P . Unless we state otherwise, we will assume that X is a reduced set of centers , that is, that B [ X ] (cid:54) = B [ X \ { x } ] for any x ∈ X .The face structure of a 3-dimensional ball polyhedron B [ X ] are defined in a naturalway: a point on the boundary of B [ X ] belonging to at least three generating spheres iscalled a vertex ; a connected component of the intersection of two generating spheres and B [ X ] is called an edge , if it is a non-degenerate circular arc; and the intersection of agenerating sphere and B [ X ] is called a face .The face structure of a ball polyhedron, unlike that of a convex polyhedron, is not neces-sarily an algebraic lattice, with respect to containment, see [BN06]. Following [BLNP07],we call a ball polyhedron in R standard , if its vertex-edge-face structure is a lattice withrespect to containment. This is the case if, and only if, the intersection of any two faces iseither empty, or one vertex or one edge, and any two edges share at most one vertex. Thepaper [KMP10] and Chapter 6 of the beautiful book [MMO19] by Martini, Montejanoand Oliveros provide further background on the theory of ball polyhedra.A fundamental result of Steinitz (see, [Zie95], [SR34] and [Ste22]) states that a graph G is the edge-graph of a 3-dimensional convex polyhedron if and only if, G is simple (ie.,it contains no loops and no parallel edges), plane and 3 -connected (ie., removing any twovertices and the edges adjacent to them yields a connected graph). In [BLNP07], it isshown that the edge-graph of any standard ball polyhedron in R is simple, plane and3-connected. Solving an open problem posed in [BLNP07] and [Bez13], our main resultshows that the converse holds as well. Theorem 1.
Every -connected, simple plane graph is the edge-graph of a standard ballpolyhedron in R . Mathematics Subject Classification.
Key words and phrases.
Steinitz’s theorem, polyhedron, ball polyhedron, edge-graph. a r X i v : . [ m a t h . M G ] N ov he proof of Steinitz’s theorem consists of two parts. First, it is shown that 3-connected, simple plane graphs can be “reduced” by a finite sequence of certain graphoperations to the complete graph K on four vertices. Second, in the geometric part, itis shown that if a graph G is obtained from another graph G (cid:48) by such an operation and G is realizable as the edge-graph of a polyhedron, then G (cid:48) is realizable as well. To proveTheorem 1, we use the first, combinatorial part without modification. Our contributionis the proof of the second, geometric part in the setting of ball polyhedra.The structure of the paper is the following. First, in Section 2, we introduce these op-erations on graphs, and recall facts on the face structure of the dual of a ball polyhedron.In Section 3, we state our main contribution, Theorem 2, which shows the “backwardinheritance” of realizability by ball polyhedra under these graph operations, and deduceTheorem 1 from it. Finally, in Section 4, we prove Theorem 2.2. Preliminaries
Simple ∆ -to- Y and Y -to- ∆ reductions on a plane graph. Let G be a 3-connected plane graph and K be a triangular face with vertices v , v and v (resp., K , be a subgraph consisting of a 3-valent vertex v of G , its neighbors v , v , v , and the edges { v, v i } connecting v to its neighbors). A ∆ Y operation is defined as the graph operationwhich removes the edges { v i , v j } of a triangular face K , adds a new vertex v from theface, and connects it to v i s, or vice versa, it takes a subgraph K , of G , removes the vertex v and the edges incident to it, then connects all pairs v − I, v j by an edge. To specifythe direction of the transformation, we will distinguish between a ∆-to- Y transformationand a Y -to-∆ transformation (see Figure 1). v v v K v v v v ⇐⇒ K , Figure 1. = ⇒ : A ∆-to- Y transformation; ⇐ =: A Y -to-∆ transformationA ∆ Y operation may create multiple edges or vertices of degree two. A graph withsuch objects is clearly not the edge-graph of a standard ball polyhedron. To fix theseissues, we define the following notion. A series-parallel reduction , or SP-reduction is thereplacement of a pair of edges incident to a vertex of degree 2 with a single edge or, thereplacement of a pair of parallel edges with a single edge that connects their commonendpoints, see Figure 2. v v = ⇒ v v v v = ⇒ v v Figure 2.
Examples of SP-reductionsAssume that a graph G contains K , as a subgraph whose degree 3 vertex is denotedby v , and its neighbors are v , v , v (resp., K with vertices v , v , v ), see Figure 1. We all edges of G that connect two vertices of K , (resp., K ) internal edges . We definethe outer degree of a neighbor of v (resp., a vertex of K ), as the number of non-internaledges adjacent to it. A K , is called Y , Y , Y , or Y if it has zero, one, two, or threeinternal edges respectively. A K is called ∆ , ∆ , ∆ , or ∆ if it has zero, one, two, orthree vertices of outer degree one, respectively.A simple ∆ Y reduction means any ∆ Y operation followed immediately by SP-reductionsthat are then possible. There are four different types of simple ∆-to- Y and Y -to-∆ re-ductions (cf. Corollary 4.7 of [Zie95]), as shown in Figure 3. Proposition 2.1.
Every -connected plane graph G can be reduced to K by a sequenceof simple ∆ Y reductions. = ⇒ = ⇒ = ⇒ = ⇒ = ⇒ = ⇒ = ⇒ ( A ) = ⇒ ( B ) Figure 3. ( A ) Four types of simple ∆-to- Y reduction, and ( B ) four typesof simple Y -to-∆ reduction, where the dotted lines denote edges that mayor may not be present, and are not affected by the simple ∆-to- Y and Y -to-∆ reductions.2.2. Standard graphs.
A planar graph with a fixed drawing on the plane is called a plane graph . It is well known that 3-connected planar graphs have only one drawing, thatis, all plane drawings of such a graph have isomorphic face lattices [Zie95, Section 4.1].
Definition 2.1.
Let G be a plane graph. We call G standard , if (i) the intersection of any two faces is either empty, or one vertex or one edge, and (ii) any two edges share at most one vertex. Remark 2.1.
Let G be the edge-graph of a ball polyhedron. Then G is standard if andonly if the ball polyhedron is a standard ball polyhedron. We leave the proof of the following two lemmas to the reader as an exercise. emma 2.1. Let G be a -connected plane graph and let the graph G be derived from G by a simple ∆ -to- Y reduction. If G is a standard graph, then so is G . The subdivision of an edge { t , t } of a graph G is another graph obtained from G byremoving the edge { t , t } , then adding a new vertex t (cid:48) and, finally, adding the edges { t , t (cid:48) } and { t (cid:48) , t } . Lemma 2.2.
Let G be a standard graph, E be a face of G and { u , u } , { u , u } be twoedges of E such that u and u are non-adjacent vertices.I. If the graph H is obtained from G by adding the edge { u , u } , then H is a standardgraph.II. If the graph H (cid:48) is obtained from G by adding the edge { u , u (cid:48) } where u (cid:48) is a newvertex subdividing the edge { u , u } , then H (cid:48) is a standard graph.III. If the graph H (cid:48)(cid:48) is obtained from G by adding the edge { u (cid:48) , u (cid:48)(cid:48) } where u (cid:48) and u (cid:48)(cid:48) are two new vertices subdividing the edges { u , u } and { u , u } respectively, then H (cid:48)(cid:48) is a standard graph. Graph duality.
We denote the dual of a plane graph G by G (cid:63) , see [Zie95, Sec-tion 4.1]. It is well known that G (cid:63) is also a plane graph, and G (cid:63) is 3-connected if andonly if, G is 3-connected.According to the following fact, simple ∆-to- Y reductions and simple Y -to-∆ reduc-tions are dual to each other, see [Zie95, Section 4.2]. Proposition 2.2.
Let G and G (cid:48) be 3-connected plane graphs. Then G (cid:48) is obtained from G by a simple ∆ -to- Y reduction if and only if, G (cid:48) (cid:63) is obtained from G (cid:63) by a simple Y -to- ∆ reduction. The dual of a ball polyhedron.
In the following, F ( B [ X ]) denotes the set offaces, and V ( B [ X ]) denotes the set of vertices of the ball polyhedron B [ X ].Let B [ X ] be a ball polyhedron in R all of whose faces contain at least three vertices.In [BN06], the dual of B [ X ] is introduced as the ball polyhedron B [ V ( B [ X ])], and abijection, called the duality mapping between B [ X ] and B [ V ( B [ X ])], is given betweenthe faces, edges and vertices of B [ X ] and B [ V ( B [ X ])], consisting of the following threemappings:(1) The vertex-face mapping is V ( B [ X ]) (cid:51) v (cid:55)→ V ∈ F ( B [ V ( B [ X ])])where V is the face of B [ V ( B [ X ])] with v as its center.(2) The face-vertex mapping is F ( B [ X ]) (cid:51) F (cid:55)→ f ∈ V ( B [ V ( B [ X ])])where f is the center of the sphere supporting the face F .(3) The edge-edge mapping is the following. Two vertices in B [ V ( B [ X ])] are connectedby an edge if and only if, the corresponding faces of B [ X ] meet in an edge.Note that every face of a standard ball polyhedron contains at least three edges. Therelationship between graph duality and duality of ball polyhedra is described below. Lemma 2.3 (Theorem 6.6.5., [Bez13]) . Let P be a standard ball polyhedron of R . Thenthe intersection P (cid:63) of the closed unit balls centered at the vertices of P is another standardball polyhedron whose face lattice is dual to that of P . Proof of Theorem 1
Our main contribution follows. heorem 2. Let G (cid:48) be a 3-connected plane graph, and let the graph G be derived from G (cid:48) by a simple Y -to- ∆ reduction. If G is the edge-graph of a standard ball polyhedron in R , then so is G (cid:48) . First, we show how Theorem 2 implies Theorem 1.
Proof of Theorem 1.
Let G be a 3-connected simple plane graph. By Proposition 2.1, thegraph G reduces to K the edge-graph of the standard ball tetrahedron by a sequence ofsimple ∆ Y reductions.Now we show that the standard ball tetrahedron can be gradually turned into a real-ization of G . Let H be the edge-graph of a standard ball polyhedron and assume that H is obtained from another edge-graph H (cid:48) by a simple ∆ Y reduction. We want to showthat H (cid:48) is realized by a standard ball polyhedron. So we need to discuss two cases: First , assume that H is obtained from H (cid:48) by a simple Y -to-∆ reduction. Then byTheorem 2, H (cid:48) is realized by a standard ball polyhedron. Second , assume that H is obtained from H (cid:48) by a simple ∆-to- Y reduction. Then byProposition 2.2, we get that the edge-graph H (cid:63) is obtained from the edge-graph H (cid:48) (cid:63) bya simple Y -to-∆ reduction. By Lemma 2.3, the edge-graph H (cid:63) is realized by a standardball polyhedron, and by Theorem 2, the edge-graph H (cid:48) (cid:63) is realized by a standard ballpolyhedron. Again by Lemma 2.3, the edge-graph H (cid:48) is realized by a standard ballpolyhedron, and this completes the proof. (cid:3) dual H H simple ∆-to- Y By assumption, H is theedge-graph of a standardball-polyhedron. H ? H ? simple Y -to-∆dual dual By Lemma 2.3, H ? is theedge-graph of a standardball-polyhedron.By Theorem 2, H ? is theedge-graph of a standardball-polyhedron.By Lemma 2.3, H is theedge-graph of a standardball-polyhedron. Proof of Theorem 2
Let ∅ (cid:54) = X ⊂ R be a finite set and B [ X ] be a standard ball polyhedron with edge-graph G , and assume that G is obtained from a graph G (cid:48) by a simple Y -to-∆ reduction.We need to show that G (cid:48) is realized by a standard ball polyhedron.Let Λ denote the triangular face of B [ X ] which realizes the triangle obtained in the Y -to-∆ reduction, let S ( x Λ ) be its supporting unit sphere, and v , v , v be the verticesof Λ and e , e , e the edges. Let F , F and F denote the faces of B [ X ] distinct from Λcontaining e , e and e respectively, and let S ( x ), S ( x ) and S ( x ) be the unit spheressupporting these faces, see Figure 4.The starting point of the proof of Theorem 2 is the removal of the ball that generatesthe triangular face Λ. Thus, we obtain another ball polyhedron, B [ X \ { x Λ } ]. Thefollowing lemma (which we prove later) describes the edge-graph of B [ X \ { x Λ } ] and,combined with Lemma 2.1 yields that it is a standard graph, and hence, by Remark 2.1, B [ X \ { x Λ } ] is a standard ball polyhedron. Lemma 4.1.
The edge-graph of the ball polyhedron B [ X \ { x Λ } ] is obtained by a simple ∆ -to- Y reduction applied to Λ in the role of K . e F e F e Λ v v v Figure 4.
The edge-graph of B [ X \ { x Λ } ] described in Lemma 4.1 may be G (cid:48) , in which case weare done. However, it may happen that this is not G (cid:48) , more precisely, the graph G isderived from G (cid:48) by a simple Y -to-∆ reduction, but the converse is not always true, itmay happen that G (cid:48) is not derived from G by a simple ∆-to- Y reduction. The reason isthat when we do a ∆-to- Y reduction, the vertices of the triangle of outer degree one in G become degree two vertices in the graph obtained from G by a ∆-to- Y reduction. Next,we do the SP -reduction, and these vertices are lost, see Figure 5 (C) and (D). Moreover,the internal edges will be missing as well, see Figure 5 (B), (C) and (D).The following lemma describes how the edge-graph of B [ X \ { x Λ } ] is converted into G (cid:48) by adding the missing vertices and edges. We achieve this by adding some extraballs. Lemma 2.2 yields that G (cid:48) is a standard graph, and hence, by Remark 2.1, the ballpolyhedron realizing G (cid:48) is a standard ball polyhedron. ( A ) = ⇒ Y -to-∆ cv v v = ⇒ ∆-to- Yv v v cv v v ( B ) = ⇒ Y -to-∆ cv v v = ⇒ ∆-to- Yv v v cv v v ( C ) = ⇒ Y -to-∆ v v c v v = ⇒ ∆-to- Yv v v v v c v v ( D ) = ⇒ Y -to-∆ v v cv v v = ⇒ ∆-to- Yv v v v v cv v v Figure 5. emma 4.2. Let ∅ (cid:54) = Z ⊂ R be a finite set and B [ Z ] be a ball polyhedron. If the edge-graph of B [ Z ] contains Y as an induced subgraph whose vertices are u , u , u and u ,and whose edges are a , a and a , see Figure 6, left side, thenI. there exists a center w such that the edge-graph of the ball polyhedron B [ Z ∪ { w } ] is obtained from the edge-graph of B [ Z ] by adding the internal edge { u , u } .II. there exists a center w (cid:48) such that the edge-graph of the ball polyhedron B [ Z ∪ { w (cid:48) } ] is obtained from the edge-graph of B [ Z ] by adding the edge { u , u (cid:48) } , where u (cid:48) isa new vertex subdividing the edge a .III. there exists a center w (cid:48)(cid:48) such that the edge-graph of the ball polyhedron B [ Z ∪{ w (cid:48)(cid:48) } ] is obtained from the edge-graph of B [ Z ] by adding the edge { u (cid:48) , u (cid:48) } , where u (cid:48) and u (cid:48) are two new vertices subdividing the edges a and a respectively. In summary, proving Lemmas 4.1 and 4.2, we prove Theorem 2, which in turn yieldsTheorem 1.4.1.
Proof of Lemma 4.1.
In this section, we use the notation of Lemma 4.1 andFigure 4.The following claim is obvious.
Claim 4.1.
For any i ∈ { , , } , Λ \ e i is contained in the interior of B [ x i ] . The following claim is the key of our proof. It states that the “new part” of theboundary of the new ball polyhedron B [ X \ { x Λ } ] belongs to the union of S ( x ), S ( x )and S ( x ). Claim 4.2. bd( B [ X (cid:48) ]) \ bd( B [ X ]) ⊆ S ( x ) ∪ S ( x ) ∪ S ( x ) , where X (cid:48) = X \ { x Λ } .Proof. Consider a point q ∈ bd( B [ X (cid:48) ]) \ bd( B [ X ]). Then there exists a generating sphere S ( x q ) of B [ X (cid:48) ] such that q ∈ S ( x q ) and x q ∈ X (cid:48) , implying that S ( x q ) is a generating sphereof B [ X ] as well. Let F = S ( x q ) ∩ B [ X ] and F (cid:48) = S ( x q ) ∩ B [ X (cid:48) ]. Then F = F (cid:48) ∩ B [ x Λ ], F ⊆ F (cid:48) , q ∈ F (cid:48) , and q / ∈ F . This yields that F (cid:48) ∩ S ( x Λ ) is a non-degenerate circular arc in F (cid:48) that separates q from F . Thus F intersects Λ in a non-degenerate circular arc. Thatonly happens if F intersects Λ in an edge of B [ X ], and hence, x q = x , x , or x . (cid:3) The following Claim is obvious.
Claim 4.3.
Let B , B and B be closed unit balls in R such that B ∩ B ∩ B is aball polyhedron with three faces. Then B ∩ B ∩ B is a ball polyedron with two verticesconnected by three edges. Finally, we are in the position to prove Lemma 4.1. By Claim 4.3, the boundary of B [ x ] ∩ B [ x ] ∩ B [ x ] contains two vertices of degree three say q and ¯ q , three edges, andthree faces. We need to prove that the “new part” N := bd( B [ X \ { x Λ } ]) \ bd( B [ X ]) ofthe boundary of the ball polyhedron B [ X \ { x Λ } ] contains either q or ¯ q with part fromeach of the three edges, and part from each of the three faces (i.e., K , ). It means thatwhen we remove the ball B [ x Λ ], the triangular face Λ of G will be replaced by Y = K , in the edge-graph of B [ X \ { x Λ } ], i.e., the edge-graph of B [ X \ { x Λ } ] is derived from G by a simple ∆-to- Y reduction.Let Γ := B [ x ] ∩ B [ x ] ∩ B [ x ] and γ := bd(Λ) = e ∪ e ∪ e , see Figure 4. By Claim 4.2, N ⊆ S ( x ) ∪ S ( x ) ∪ S ( x ), this implies that N ⊆ bd(Γ). Clearly, bd(Γ) has three edges,say e (cid:48) , e (cid:48) and e (cid:48) such that e (cid:48) ⊆ S ( x ) ∩ S ( x ), e (cid:48) ⊆ S ( x ) ∩ S ( x ) and e (cid:48) ⊆ S ( x ) ∩ S ( x ).By Claim 4.1, v i ∈ int ( B [ x i ]) for all i = 1 , ,
3. By Claim 4.3, S ( x ) ∩ S ( x ) ∩ S ( x ) isa set of two points q and ¯ q . Clearly, { v , v , v } ∩ { q, ¯ q } = ∅ . ince e (cid:48) \ { q, ¯ q } ⊆ int( B [ x ]), v ∈ e (cid:48) \ { q, ¯ q } and { e (cid:48) , e (cid:48) } ⊆ S ( x ), this implies that v belongs to e (cid:48) \ { q, ¯ q } and does not belong to e (cid:48) or to e (cid:48) . Similarly, v i belongs to e (cid:48) i \ { q, ¯ q } ( i = 2 ,
3) only, respectively. Thus, exactly one of v , v and v is contained oneach of the three edges of Γ.Observe that both S ( x Λ ) ∩ Γ and Λ are the intersections of three spherical disks on S ( x Λ ), each smaller than a hemi-sphere: S ( x Λ ) ∩ B [ x ], S ( x Λ ) ∩ B [ x ] and S ( x Λ ) ∩ B [ x ].Hence, S ( x Λ ) ∩ Γ = Λ, and it follows that S ( x Λ ) ∩ bd(Γ) = γ .It follows that γ partitions bd(Γ) into two components, q is in one component and ¯ q is in the other, we may assume that ¯ q ∈ int ( B [ x Λ ]). Claim 4.2 yields that q ∈ N asrequired. This completes the proof of Lemma 4.1.4.2. Proof of Lemma 4.2. (I) Let g , g and g be the faces of B [ Z ] such that a = g ∩ g , a = g ∩ g and a = g ∩ g and let S ( z ), S ( z ) and S ( z ) be the spheres supporting these faces, seeFigure 6, left side.To add an internal edge to Y , we will add a rotated copy z (cid:48) of z to the set Z , where theaxis of the rotation is the line through u and u , the angle of the rotation is sufficientlysmall, and u is outside of B [ z (cid:48) ]. Thus, we obtain a new triangular face g (cid:48) supported by S ( z (cid:48) ), see the dashed lines on Figure 6 ( A ), and remove the dotted lines on Figure 6 ( A ). ( A ) = ⇒ g g g a a a uu u u g g g g uu u u ( B ) = ⇒ g g g a a a uu u u u g g g g uu u u ( C ) = ⇒ g g g a a a uu u u g g g g u u uu u u Figure 6.
The dotted (blue) vertices and edges are removed and thedashed (red) vertices and edges are introduced in the new graph.(II) To add a new vertex and a new edge to Y , we use the same method as in (I),but we choose the rotation axis so that it passes through the vertex u and intersects theedge a at a point, say u (cid:48) , distinct from its endpoints, see Figure 6 ( B ).(III) To add two new vertices and a new edge to Y , we use again the same method of(I), but this time, we choose the rotation axis so that it intersects the edges a and a at wo points, say u (cid:48) and u (cid:48) respectively, distinct from their endpoints, see Figure 6 ( C ).This finishes the proof of Lemma 4.2. Acknowledgements
SMA would like to thank the Tempus Public Foundation (TPF), Stipendium Hungar-icum program, and University of Thi-Qar, Iraq for the support for his PhD scholarship.ZL was supported by grants K119670 and BME Water Sciences & Disaster PreventionTKP2020 IE of the National Research, Development and Innovation Fund (NRDI), bythe ´UNKP-20-5 New National Excellence Program of the Ministry for Innovation andTechnology, and the J´anos Bolyai Scholarship of the Hungarian Academy of Sciences.MN was supported by the National Research, Development and Innovation Fund(NRDI) grant K119670, by the ´UNKP-20-5 New National Excellence Program of theMinistry for Innovation and Technology from the source of the NRDI, as well as theJ´anos Bolyai Scholarship of the Hungarian Academy of Sciences.
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SAMI MEZAL ALMOHAMMAD: INST. OF MATH., LOR ´AND E ¨OTV ¨OS UNIV., BU-DAPEST, HUNGARY.DEPT. OF MATH., FACULTY OF COMP. SCI. AND MATH., UNIV. OF THI-QAR, THI-QAR, IRAQ.
Email address : [email protected], [email protected]
ZSOLT L ´ANGI: MTA-BME MORPHODYNAMICS RESEARCH GROUP AND DEPART-MENT OF GEOMETRY, BUDAPEST UNIVERSITY OF TECHNOLOGY, BUDAPEST, HUN-GARY
Email address : [email protected]
M ´ARTON NASZ ´ODI: ALFR´ED R´ENYI INST. OF MATH.; MTA-ELTE LEND ¨ULET COM-BINATORIAL GEOMETRY RESEARCH GROUP; DEPT. OF GEOMETRY, LOR ´AND E ¨OTV ¨OSUNIVERSITY, BUDAPEST