LLOCAL CERTIFICATION OF GRAPHS ON SURFACES
LOUIS ESPERET AND BENJAMIN L´EVˆEQUE
Abstract.
A proof labelling scheme for a graph class C is an assignment of certificatesto the vertices of any graph in the class C , such that upon reading its certificate andthe certificate of its neighbors, every vertex from a graph G ∈ C accepts the instance,while if G (cid:54)∈ C , for every possible assignment of certificates, at least one vertex rejectsthe instance. It was proved recently that for any fixed surface Σ, the class of graphsembeddable in Σ has a proof labelling scheme in which each vertex of an n -vertex graphreceives a certificate of at most O (log n ) bits. The proof is quite long and intricate andheavily relies on an earlier result for planar graphs. Here we give a very short prooffor any surface. The main idea is to encode a rotation system locally, together with aspanning tree supporting the local computation of the genus via Euler’s formula. Introduction
The goal of local certification is to verify that a network, represented by a connectedgraph G in which each vertex has a unique identifier, satisfies some given property. Theconstraint is that each node of the network as a local view of the network (its neighborhood)and has to make its decision based only on this local view. If the graph satisfies theproperty, we want all vertices to accept the instance, while if the graph does not satisfy theproperty, at least one vertex has to reject the instance. This is a significant restriction andit only allows the verification of local properties (related to the degrees, for instance), soeach vertex is given in addition some small certificate, which can be seen as an advice, andeach vertex can now base its decision on its certificate and the certificates of its neighbors.For any property P , the goal is to produce a protocol to certify P locally while usingcertificates of minimal size. Such a protocol is called a one-round proof labelling schemewith complexity f ( n ), where f ( n ) is the maximum number of bits in the certificate of avertex in an n -vertex graph satisfying P (a formal definition of proof labelling schemeswill be given in Section 2). Proof labelling schemes appear naturally in self-stabilizingprotocols, and are particular form of distributed interactive protocols.For any integer g ≥
0, let C g be the class of graphs embeddable on a surface of Eulergenus at most g . In particular C is the class of planar graphs. We also let O g be theclass of graphs embeddable on an orientable surface of Euler genus at most g , and N g bethe class of graphs embeddable on an non-orientable surface of Euler genus at most g (seeSection 3 for more on surfaces and graph embeddings).Motivated by recent work on distributed interactive protocols in classes with lineartime recognition algorithms [9], it was recently proved that planar graphs have a one-round proof labelling scheme with complexity O (log n ) [3], and that this complexity isbest possible. More recently, the same authors built upon their previous work to extendtheir result to graphs embeddable on any fixed surface [4]. Theorem 1.1 ([4]) . For any integer g ≥ , the classes C g , O g , and N g have a one-roundproof labelling scheme with complexity at most O ( √ g · log n ) . The proof of the planar case in [3] and its extension to general surfaces in [4] are fairlyintricate, with the two papers totaling 65 pages. In this short note, we give a simple a r X i v : . [ m a t h . C O ] F e b L. ESPERET AND B. L´EVˆEQUE and direct proof of Theorem 1.1, based on rotation systems together with a distributedcomputation of the Euler genus using Euler’s formula along a rooted spanning tree.The formal definition of proof labelling scheme is given in Section 2, and the basicterminology of graphs on surfaces is given in Section 3, along with a description of rotationsystems and the Heffter-Edmonds-Ringel rotation principle. For the description of thecertificates, we found it convenient to first present the orientable case (Section 4), whichis slightly simpler, and then explain the small modifications we have to perform in thenon-orientable case (Section 5). We could have presented everything in the latter setting,which is more general, but we believe it would not have helped the reader.2.
Proof labelling schemes A one-round proof labelling scheme for a graph class F is a pair ( P , V ), such thatfor any integer n ≥ n -vertex graph G , whose vertices are assigned distinctidentifiers from { , . . . , poly( n ) } , the prover P assigns to each vertex v ∈ V ( G ) a certificate c G ( v ) ∈ { , } ∗ (that might depend on the vertex identifiers), so that the verifier V satisfiesthe following properties: • each vertex v ∈ V ( G ) collects the certificates C G ( v ) = { c G ( u ) | uv ∈ E ( G ) } of itsneighbors ( one-round ) • for any graph G ∈ F and any vertex v ∈ V ( G ), V ( c G ( v ) , C G ( v )) = 1 ( completeness ) • for any graph G (cid:54)∈ F , there exists a vertex v ∈ V ( G ) such that V ( c G ( v ) , C G ( v )) = 0( soundness ).In other words, upon reading its certificate and the certificates of its neighbors, allvertices of a graph G ∈ F accept the instance, while at least one vertex in each graph G (cid:54)∈ F rejects the instance.The complexity of the labelling scheme is the maximum size | c G ( v ) | of a certificate inan n -vertex graph of F . If we say that the complexity is O ( f ( n )), for some function f ,the O ( · ) notation refers to n → ∞ . The definition above assumes that there is a singleround of communication between the vertices (when each node collects the certificates ofits neighbors), this is why this type of proof labelling scheme is called a one-round prooflabelling scheme. There is a more general definition [7] in which each node is allowed togather the certificates of its neighbors at distance t , for some integer t ≥
1, but in our caseit is enough to restrict ourselves to t = 1.3. Cellular embeddings and rotation systems
Surfaces.
We refer the reader to the book by Mohar and Thomassen [8] for moredetails or any notion not defined here. All the graphs in this paper are simple (i.e. withoutloops and multiple edges), and connected. A surface is a non-null compact connected 2-manifold without boundary. A surface can be orientable or non-orientable. The orientablesurface of genus h is obtained by adding h ≥ handles to the sphere, while the non-orientable surface of genus k is formed by adding k ≥ cross-caps to the sphere. The Euler genus of a surface Σ is defined as twice its genus if Σ is orientable, and as its(non-orientable) genus otherwise.We say that an embedding is cellular if every face is homeomorphic to an open diskof R . If G is a graph with a cellular embedding in a surface of Euler genus g , then Euler’sformula states that | V ( G ) | − | E ( G ) | + | F ( G ) | = 2 − g, where V ( G ), E ( G ), and F ( G ) denote the set of vertices, edges, and faces of (the embeddingof) G .A graph G is k -degenerate if there is an ordering v , . . . , v n of the vertices of G , suchthat for any 1 ≤ i ≤ n , the vertex v i has at most k neighbors v j with j < i . Using Euler’s OCAL CERTIFICATION OF GRAPHS ON SURFACES 3 formula, it is not difficult to derive the following result due to Heawood (see [8, Theorem8.3.1]).
Theorem 3.1 (Heawood) . For every g ≥ , every graph embeddable on a surface of Eulergenus at most g is k -degenerate, with k = max (cid:16) , √ g (cid:17) . Rotation systems.
Let G be a graph. A half-edge of G is a pair ( v, e ), where v ∈ V ( G ) and e is an edge incident to v . We say that ( v, e ) and v are incident. The set ofall half-edges of G is denoted by B ( G ). A rotation system of G is a pair of permutations( σ, α ) acting on B ( G ), such that • for any edge e = uv ∈ E ( G ), α ( v, e ) = ( u, e ) and α ( u, e ) = ( v, e ) (i.e., α is aninvolution with no fixed point), and • for each orbit of σ , there is a vertex v ∈ V ( G ) such that the orbit consists of allthe half-edges incident to v (in other words, we can view σ as a circular order onthe half-edges incident to each vertex of G ).Each cellular embedding of a graph G in some orientable surface Σ can be translatedinto a rotation system by defining α as above and σ as the collection of circular orderson the half-edges around each vertex, in the positive orientation of the surface. Note thateach orbit of α ◦ σ corresponds to a different face of the embedding (where the half-edgesappear in the negative orientation of the surface). ασ σ ◦ α Figure 1.
The conversion of a graph embedded in the plane (or thesphere) into a rotation system, and back into a graph embedded in thesphere (obtained by gluing the two polygons on edges of matching colors).The Heffter-Edmonds-Ringel rotation principle (see Theorem 3.2.4 in [8]) states thatevery cellular embedding of a connected graph G in an orientable surface is uniquelydetermined, up to homeomorphism, by its rotation system.Although it will not be needed in the remainder, it is worth explaining how to recoverthe embedding of G in Σ from the rotation system ( σ, α ) with ground set B . Each orbit σ v of σ is associated to a distinct vertex v , and each orbit α e of α is associated to adistinct edge e connecting the two vertices associated to the two elements of α e . Theresulting graph is precisely G . To each orbit of α ◦ σ , we associate a polygon f whosesides are indexed by the edges of G and whose vertices are indexed by the vertices of G .Note that vertices and edges of G might appear several times on the same polygon or ondifferent polygons (indeed, each edge appears twice and each vertex v appears d ( v ) timesamong all the polygons). The circular order on the vertices and edges on each polygonin the negative orientation coincides with the circular order of the elements of B in thecorresponding orbit of α ◦ σ . For any edge e of G , we glue the two polygons containing e together on e (if a single polygon contains e twice, we glue the two sides corresponding to e together), by respecting the natural orientation of e (that is, if e = uv , the vertex u ofone polygon is identified with the vertex u of the other polygon, and similarly for v ), seeFigure 1. L. ESPERET AND B. L´EVˆEQUE The certificate
We now describe our certificate for being embeddable on an orientable surface of Eulergenus at most g , as well as the verification part at each vertex. Recall that all graphs inthis paper are assumed to be simple and connected.Let G be a connected graph embeddable on an orientable surface of Euler genus at most g . By considering an embedding of G on an orientable surface of minimum Euler genus,we can assume that G has a cellular embedding in an orientable surface Σ of Euler genusat most g (see [8, Propositions 3.4.1]). Consider such a cellular embedding of G in Σ,and let F ( G ) denote the set of faces of the embedding. Let ( σ, α ) be the rotation systemassociated to the cellular embedding of G in Σ.Our certificate consists in two parts: (1) a (local) description of the rotation system( σ, α ), and (2) a spanning tree of G which supports the (local) computation of the Eulergenus, via Euler’s formula.4.1. A distributed rotation system.
Let v ∈ V ( G ). The certificate of v contains theidentifier id( v ) of v . Since for any vertex u in G , id( u ) ∈ { , . . . , poly( n ) } , storing aconstant number of identifiers at v takes O (log n ) bits.Consider a half-edge ( v, e ). The face f associated to the orbit of σ ◦ α containing ( v, e )is said to be the face bounding the half-edge ( v, e ), and we say that ( v, e ) is bounded by f . The half-edge ( σ ◦ α )( v, e ) is called the next half-edge on f with respect to ( v, e ). Notethat if an edge e of G is incident to a single face f , the two half-edges of e are boundedby f , while if e is incident to two distinct faces f , f , one half-edge of e is bounded by f and the other is bounded by f .In the remainder, it will be convenient to talk about the identifier id(( v, e )) of a half-edge ( v, e ), which we define as the the pair (id( v ) , id( u )), where u is the endpoint of e distinct from v .For each face f of G , the prover considers an arbitrary half-edge bounded by f andsets it as the root of f (in the remainder, if the root of f is ( v, e ), we say that f pointsto v ). The prover then assigns integers to the half-edges bounded by f as follows: for anyhalf-edge ( v, e ) bounded by f , the f -index of ( v, e ) is the smallest integer i ≥ v, e ) = ( σ ◦ α ) i ( u, e ), where ( u, e ) denotes the root half-edge of f . So the root half-edgeof f has f -index 0, and the maximum f -index is d ( f ) −
1, where d ( f ) denotes the degreeof f (the number of edges in a boundary walk of f , where edges appearing twice in thewalk are counted with repetition). Note that if some half-edge has f -index i , the nexthalf-edge on f has f -index i + 1 if and only if it is different from the root half-edge of f .Our main goal is that after having collected the certificates of all its neighbors, • v knows the orbit of σ associated to v (i.e. the circular ordering of the neighborsof v associated to the embedding), • for each edge e incident to v , the vertex v knows whether ( v, e ) is the root half-edgeof the face bounding ( v, e ).Note that we cannot store the circular ordering of the neighbors of v explicitly in thecertificate of v , as it would require Ω(∆ log n ) bits for a vertex of degree ∆ (which can beas large as n − v while maintainingthe property that each vertex stores a certificate of at most O (log n ) bits (this idea wasalso used in [3, 4]).By Theorem 3.1, there is an order v , . . . , v n on the vertices of G , such that each vertex v i has at most k = O ( √ g ) neighbors v j with j < i . The neighbors v j of v i with j < i OCAL CERTIFICATION OF GRAPHS ON SURFACES 5 are called the special neighbors of v i . Note that for any edge uv , exactly one of u, v is aspecial neighbor of the other.For each vertex v , let ( v, e ) , ( v, e ) , . . . , ( v, e d ( v ) − ) be the half-edges incident to v inthe positive orientation, starting with some arbitrary half-edge ( v, e ) incident to v . Foreach 0 ≤ i ≤ d ( v ) −
1, we say that ( v, e i ) has v -index i , where v -indices are understoodmodulo d ( v ), and we denote this half-edge by (cid:104) v (cid:105) i . By extension, we also say that theendpoint of e i distinct from v has v -index i , so that the circular ordering of the half-edgesaround v coincides with a circular ordering of the neighbors of v (this is possible here sincewe deal with simple graphs).Recall that each vertex v stores its identifier id( v ). In addition, for each special neighbor u of v , the vertex v stores the identifier id( u ) followed by: • the u -index i of v and the v -index j of u , • the identifier of the root half-edge of the face f bounding the half-edge (cid:104) u (cid:105) i , andthe f -index of (cid:104) u (cid:105) i . • the identifier of the root half-edge of the face f (cid:48) bounding the half-edge (cid:104) v (cid:105) j , andthe f (cid:48) -index of (cid:104) v (cid:105) j .Each vertex has degree at most n − | E ( G ) | = O ( n ),so storing each v -index or f -index takes at most O (log n ) bits. It follows that storing allthe information described above takes at most O ( k log n ) bits per vertices.The verification process is the following. Each vertex collects the certificates of all itsneighbors. For each edge uv , either u is a special neighbor of v or v is a special neighbor of u . So after having collected the certificates of their neighbors, both u and v are supposedto have all the information concerning the edge uv , namely: the identifiers id( u ) and id( v ),the u -index i of v , the v -index j of u , the identifiers of the root half-edges of the faces f and f (cid:48) bounding (cid:104) u (cid:105) i and (cid:104) v (cid:105) j respectively, the f -index of (cid:104) u (cid:105) i and the f (cid:48) -index of (cid:104) v (cid:105) j . Theverifier at each vertex v checks that for each neighbor u of v , the information concerningthe edge uv was either received from u or stored at v , and not both (so u and v haveconsistent information concerning the edge uv ). The verifier at v then checks that the setof v -indices of the neighbors of v form a circular permutation of { , . . . , d ( v ) − } , thuscertifying that the information collected by v is consistent with the local view of v in someembedding of G .Let ( σ, α ) be the rotation system given by the v -indices of incident half-edges at eachvertex v . By the Heffter-Edmonds-Ringel rotation principle, ( σ, α ) defines a unique cellularembedding of G on an orientable surface Σ (up to homeomorphism). Note that for anyedge uv where i is the u -index of v and j is the v -index of u , and f is the face bounding (cid:104) u (cid:105) i in Σ, the next half-edge on f with respect to (cid:104) u (cid:105) i is (cid:104) v (cid:105) j +1 (see Figure 2). u (cid:104) v (cid:105) j (cid:104) u (cid:105) i (cid:104) u (cid:105) i +1 (cid:104) v (cid:105) j +1 f v Figure 2.
The next edge on f . L. ESPERET AND B. L´EVˆEQUE
We now proceed to the ”verification of faces”. For each vertex v , we do the following.For any 0 ≤ j ≤ d ( v ) −
1, let u j be the neighbor of v with v -index j , and let π ( j ) be the u j -index of v . For any 0 ≤ j ≤ d ( v ) −
1, the verifier at v checks that the half-edges (cid:104) u j (cid:105) π ( j ) and (cid:104) v (cid:105) j +1 agree on the identifier of the root half-edge of the face f bounding them, sothat the knowledge of the root half-edge of f is consistent along the face f . This showsthat each face has a unique root half-edge. In order to make sure that this root half-edgeof f is actually bounded by f , the verifier at v simply checks that for 0 ≤ j ≤ d ( v ) − f -index of (cid:104) v (cid:105) j +1 is equal to 0 if (cid:104) v (cid:105) j +1 is the root half-edge of f , or equal to 1 plusthe f -index of (cid:104) u j (cid:105) π ( j ) otherwise. Since the face f is finite and circularly ordered, somehalf-edge ( u, e ) must have f -index at least the f -index of the next half-edge ( u (cid:48) , e (cid:48) ) on f ,and by definition this is only possible if ( u (cid:48) , e (cid:48) ) has f -index 0. It follows that if the verifierat each vertex agrees with the instance, each face f has unique root half-edge, and thisroot half-edge is bounded by f (so f points to a unique vertex, and this vertex is lying on f ). Hence, if no vertex has rejected the instance so far, each vertex knows whether eachof its half-edges is the root half-edge of the face bounding it. In particular, each vertexknows the number of faces pointing to it.4.2. Computation of the Euler genus.
Using the information collected by each vertex v , and assuming all vertices have accepted the instance so far, we now certify that thesurface Σ has Euler genus at most g . To do this, it suffices to compute | V ( G ) | , | E ( G ) | ,and | F ( G ) | and apply Euler’s formula. We will do it by collecting the number of vertices,edges and faces along a spanning tree. Let T be a rooted spanning tree in G with root r .This spanning tree is certified locally using the following classical scheme (see [1, 2, 6]):the prover gives the identifier id( r ) of the root of T to each vertex v of G , as well as d T ( v, r ), its distance to r in T , and each vertex v distinct from the root is also given theidentifier of its parent p ( v ) in T . The verifier at v starts by checking that v agrees withall its neighbors in G with the identity of the root r of T . If so, if v (cid:54) = r , v checks that d T ( v, r ) = d T ( p ( v ) , r ) + 1. Once the rooted spanning tree T has been certified, each vertexof G knows its children in T . This can be used to check that • all vertices agree on the same number n = | V ( G ) | of vertices: In order to do this,the prover gives n to each vertex v of G , as well as a counter ν ( v ) which is equalto the number of vertices in the subtree of T rooted in v . The verifier at everyvertex v simply checks that v has the same value of n as its neighbors in G , andthat ν ( v ) is equal to 1 plus the sum of ν ( u ), for all children u of v (if any). Notethat this can be checked locally. It only remains to check that for the root r of T , ν ( r ) = n . • all vertices agree on the same number m = | E ( G ) | of edges: Again, the provergives the value of m to each vertex v of G , together with a counter µ ( v ) definedas the half of the sum of the degrees of the vertices in the subtree of T rooted in v . The verifier at v only needs to check that it agrees on the value of m with itsneighbors in G , and that µ ( v ) is d G ( v ) plus the sum of µ ( u ), for all children u of v (if any). Since m = | E ( G ) | = (cid:80) v ∈ V ( G ) d G ( v ), it remains to check that for theroot r of T , µ ( r ) = m . • all vertices agree on the same number | F ( G ) | of edges: Again, the prover gives thevalue of | F ( G ) | to each vertex v of G , together with a counter φ ( v ) equal to thenumber of faces pointing to vertices lying in the subtree of T rooted in v (recallthat each face has a unique root half-edge, and each vertex knows the number offaces pointing to it). The verifier at v checks that φ ( v ) is the number of facespointing to v plus the sum of φ ( u ), for all children u of v (if any). It remains tocheck that for the root r of T , φ ( r ) = | F ( G ) | . OCAL CERTIFICATION OF GRAPHS ON SURFACES 7
It follows that, assuming no vertex has rejected the instance so far, each vertex has nowaccess to | V ( G ) | , | E ( G ) | , and | F ( G ) | , and can check whether2 + | E ( G ) | − | V ( G ) | − | F ( G ) | ≤ g. As a consequence of Euler’s formula, this is equivalent to say that the rotation systemassociated to G embeds G in an orientable surface of Euler genus at most g .5. Non-orientable surfaces
The case of non-orientable surfaces is very similar to the case of orientable surfaces,but there is an additional twist. An embedding scheme is a rotation system ( σ, α ), exceptthat each orbit e of α has a sign λ e ∈ {− , } . Given a cellular embedding of a graph G in a surface Σ (which is orientable or non-orientable), we can associate a circular orderaround the half-edges incident to each vertex v , by chosing an arbitrary orientation of thetopological neighborhood of v (positive or negative). This choice of local orders aroundthe vertices gives σ , and the edges give α , as before. Since we have chosen arbitraryorientations around the vertices, the orientations around two adjacent vertices u and v may not be consistent (i.e. agree on a small topological neighborhood around the edge uv ). If they are consistent we set λ uv = 1 and otherwise we set λ uv = −
1. The surface isorientable if and only if there is a choice of local orientations that is globally consistent,that is such that the resulting signs satisfy λ e = 1 for every edge e . σ ασ ◦ ασ ◦ α σ ασ − ◦ ασ − ◦ αλ = 1 λ = − Figure 3.
Description of the ”next” half-edge on a face depending whetherthe local orderings of two adjacent vertices are consistent (left) or not(right).It turns out that the Heffter-Edmonds-Ringel rotation principle still holds in this moregeneral setting (see Theorem 3.1.1 in [8]). The only difference when retrieving the facesof the embedding is that in order to find the ”next” half-edge on some face, with respectto some half-edge ( v, e ), we consider the sign λ e . If λ e = 1, the next half-edge on the facebounding ( v, e ) is ( σ ◦ α )(( v, e )), as before. If λ e = −
1, the next half-edge on the facebounding ( v, e ) is ( σ − ◦ α )(( v, e )). So instead of identifying the faces in the embedding of G with orbits of σ ◦ α as before, we identify them with orbits of the function ϕ : ( v, e ) (cid:55)→ ( σ λ e ◦ α )(( v, e )). This is illustrated in Figure 3.To adapt the certificate of the orientable setting to this more general framework, itsuffices to add to the certificate of each vertex v , for each special neighbor u of v , thevalue of λ uv . Using this additional information, the next half-edge on a face is computedusing ϕ instead of σ ◦ α . This adds at most k = O ( √ g ) bits to the size of the certificates,so the complexity of the proof labelling scheme remains O ( √ g log n ). References [1] Y. Afek, S. Kutten, and M. Yung,
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The power of distributed verifiers in interactive proofs , In: 31stACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1096–115, 2020.(L. Esperet) Laboratoire G-SCOP (CNRS, Univ. Grenoble Alpes), Grenoble, France
Email address : [email protected] (B. L´evˆeque) Laboratoire G-SCOP (CNRS, Univ. Grenoble Alpes), Grenoble, France Email address ::