Fixed-point logics on planar graphs

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1 Fixed-point logics on planar graphs Martin Grohe Institut für mathematische Logik Albert-Ludwigs-Universität Freiburg Eckerstr. 1, Freiburg, Germany Abstract We study the expressive power of inflationary fixed-point logic IFP and inflationary fixed-point logic with counting IFP+C on planar graphs. We prove the following results: (1) IFP captures polynomial time on 3-connected planar graphs, and IFP+C captures polynomial time on arbitrary planar graphs. (2) Planar graphs can be characterized up to isomorphism in a logic with finitely many variables and counting. This answers a question of Immerman [7]. (3) The class of planar graphs is definable in IFP. This answers a question of Dawar and Grädel [16]. 1. Introduction The basic problem of descriptive complexity theory is to find a logic L for a given complexity class K such that an isomorphism invariant property of finite structures can be defined in L if, and only if, it belongs to K. If this is the case, we say that L captures K. A restricted version of this problem is to capture a complexity class K on a class C of structures. In this case only properties of structures in C have to be considered. It is comparatively easy to capture complexity classes on the class of ordered structures; for all common complexity classes logics capturing them on the class of ordered structures are known. For classes above NP these results can usually be extended to arbitrary structures, but this is no longer the case for the classes below. In particular, inflationary fixed-point logic IFP captures P on the class of ordered structures (this result, henceforth referred to as the ILV-Theorem, has independently proved by Immerman [6], Livchak [10], and Vardi [15]), but not on arbitrary structures. The question of whether there exists a reasonable logic that captures P on arbitrary structures remains one of the most challenging in finite model theory. Some partial results are known besides the ILV-Theorem. In particular, Immerman and Lander [8] and Lindell [9] proved that the counting-extension IFP+C of IFP, introduced by Immerman [7], captures P on the class of trees. We study the problem of capturing P on the class of planar graphs. From the fact (essentially due to Hopcroft and Tarjan [4]) that there is a P-canonization-algorithm for planar graphs it follows that there is a logic that captures P on planar graphs. However, this logic is quite abstract and unsatisfying from a logicians perspective. We prove that the ILV-Theorem extends to 3-connected planar graphs, and that the Immerman-Lander-Lindell result on trees extends to arbitrary planar graphs. Theorem 1.1 (1) IFP captures P on the class of 3-connected planar graphs. (2) IFP+C captures P on the class of planar graphs. We show that we can define an ordering on 3-connected planar graphs in IFP with 3 parameters. Then we can apply the ILV-Theorem to prove (1). The extension to arbitrary planar graphs is obtained via the well-known decompositions of connected graphs into their 2-connected components and of 2-connected graphs into their 3-connected components. However, this extension requires some effort. We introduce a new notion of definable canonization for a class C of structures, which says that an ordered copy of each structure in C can be defined within the structure by means of a syntactical interpretation. This does not mean that an order is definable on the structures in C, but it still suffices to apply the ILV-Theorem and thus to prove that if a class C admits definable canonization, say, in the logic IFP+C, then IFP+C captures P on C. For example, the class of trees admits IFP+C-definable canonization. The next step is to introduce the block-decomposition of a structure, which is a straightforward generalization of a tree-decomposition of a graph. We prove a central lemma saying that if there is an

2 IFP+C-definable block decomposition for a class of structures such that the blocks admit IFP+C-definable canonization, then the class admits IFP+C-definable canonization. The reason I mention this technical lemma in the introduction is that I expect it to be useful in other contexts, too. Actually, this part of the paper has nothing to do with planar graphs. But applied to planar graphs with their decompositions mentioned above, our lemma yields IFP+C-definable canonization for the class of planar graphs and thus part (2) of the theorem. Our results have some further consequences. Immerman [7] asked how many variables are needed to characterize planar graphs up to isomorphism in first-order logic with counting quantifiers. Although we cannot give the precise number of variables, it follows easily from the IFP+Cdefinable canonization for planar graphs that a finite number of variables suffices. Let C k be the logic formed by starting with atomic formulas over a fixed set x 1 ; : : : ; x k of variables, and by deriving new formulas by means of negation, disjunction, and counting quantifiers 9 m x i, for m 2 N; i k (saying that there exists at least m elements x i ). Theorem 1.2 There is a k 1 such that for all planar graphs G there is a C k -formula ' G that characterizes G up to isomorphism. A third Theorem answers a question of Grädel and Dawar [16]: Theorem 1.3 The class of planar graphs is definable in the logic IFP. This follows from the facts that a graph is planar if, and only if, its 3-connected components are, and that planarity can be decided in P together with our results that IFP captures P on 3-connected planar graphs and that the 3-connected components of a graph are definable in IFP. Due to space limitations, many proofs in this extended abstract, in particular in Sections 3 and 4, can only be sketched. 2. Preliminaries I assume a basic knowledge in mathematical logic and complexity theory. In this paper, structures are always finite of a relational vocabulary. Classes of structures are always closed under isomorphism. The universe of a structure A is denoted by A, and the interpretation of a relation symbol R in A by R A. The expansion of a structure A by a relation is denoted by (A; ). An ordered structure is a structure whose vocabulary contains the symbol which interpreted as a linear order of the universe. For a structure A (with universe A) and a subset B A we let hbi A denote the (induced) substructure of A with universe B. Furthermore, we let A n B be ha n Bi A. We are mainly interested in graphs. We consider them as relational structures G = (G; E G ), where E G is a symmetric and anti-reflexive binary relation on the set G. We consider the edges of a graph as undirected pairs and denote them by fa; bg. We consider complexity classes as classes of languages over the alphabet f0; 1g. With each graph we can associate a set of adjacency matrices, one for each ordering of the vertices of the graph. Of course we can consider the matrices as words over f0; 1g and have thus associated a language L(G) f0; 1g with each graph G. With each class C of graphs we associate the language L(C) = S G2C L(G). Similarly, for every vocabulary we can associate a language L(A) with each -structure A and a language L(C) with each class of -structures. Definition 2.1 Let L be a logic, K a complexity class and C a class of structures. We say that L captures K on C if for each class D C we have: D is definable in L if, and only if, L(D) 2 K. The set of IFP-formulas is obtained adding the following formula-formation rule to the usual rules to form first-order formulas: Given a formula ', and for some k 1 a k-tuple x of individual variables, a k-tuple t of terms, and a k-ary relation variable X, we may form the new formula [IFP x;x ']t: To define the semantics of inflationary fixed-point logic IFP, for each IFP-formula '(x; X) and structure A we define a sequence (X A i ) by letting XA 0 = ; and X A i+1 = X A i [ fa 2 A j A j= (a; X A i )g. We let XA 1 = S i0 XA i. Then A j= [IFP x;x ']t () t A 2 X A 1 ; where t A denotes the interpretation of the term-tuple t in A. The following ILV-Theorem has independently been proved by Immerman [5], Livchak [10], and Vardi [15], in slightly different formulations. The version of the result we state here goes back to Gurevich [3]. Theorem 2.2 IFP captures P on the class of ordered structures. Our results are based on the following corollary. Let D be a class of structures and '(x; y; z 1 ; : : : ; z k ) a formula with all free variables among x; y; z 1 ; : : : ; z k. We say that '(x; y; z 1 ; : : : ; z k ) defines an ordering on D (with parameters z 1 ; : : : ; z k ), if for each structure A 2 D there are

3 c 1 ; : : : ; c k 2 A such that the binary relation fab 2 A 2 j A j= '(a; b; c 1 ; : : : ; c k )g is a linear order on A. F 12 v P P 2 1 P 3 F 12 v P P 2 1 P 3 Corollary 2.3 Let D be a class of structures such that there is an IFP-formula that defines an ordering on D, possibly with parameters. Then IFP captures P on D. F 13 F 23 x P F 13 F 23 4 y 3. IFP on 3-connected planar graphs 3.1. Topological prerequisites We review some basic definitions and facts on planar graphs. For details, the reader may consult a book on graph theory, such as [2]. A simple curve in the plane R 2 is the image of a continuous injective function g : [0; 1]! R 2. Its endpoints are g(0) and g(1), and its interior is the set of all other points. The interior of the simple curve is denoted by. A drawing of a graph G = (G; E G ) is a mapping on G [ E G that associates a point of the plane R 2 with each vertex of G and a simple curve with each edge e 2 E G in such a way that for distinct v; w 2 G we have (v) 6= (f )= ;, and (w), for distinct e; f 2 E we have (e) \ for vertices v 2 G and edges e 2 E G the point (v) is an endpoint of (e) if v is incident with e and not contained in (e) otherwise. For convenience, we let (G) = (G) [ S e2eg (e). A graph G is planar if it has a drawing. Let G be a graph and a drawing of G. A face of G under is a connected component of R 2 n (G). The boundary subgraph of a face F R 2 is the subgraph H G consisting of all vertices and edges of G whose image is contained in the boundary of F in R 2. The following lemma is essentially the Jordan curve theorem. Lemma 3.1 Let the graph C be a cycle and a drawing of C. Then C has precisely two faces under. Their boundary subgraph is C. Two paths in a graph are internally disjoint if they do not have any vertices except maybe their two end-vertices in common. Figure 1 illustrates the following Lemma. Lemma 3.2 Let G be a graph and a drawing of G. Let H be a subgraph of G consisting of three pairwise internally disjoint paths P 1 ; P 2 ; P 3 that have the same endpoints v and w. Then H has precisely three faces F 12 ; F 13 ; F 23 under (restricted to H) whose boundary-subgraphs are P 1 [ P 2, P 1 [ P 3, and P 2 [ P 3, respectively. Furthermore, if P 4 is another path in G that connects a vertex x 2 P 1 n fv; wg with a vertex y 2 P 2 n fv; wg such that F 12 \ (P 4 ) = ;, then P 4 \ P 3 6= ;. w Figure 1. The two statements of Lemma 3.2 Recall that a graph G is k-connected, for a k 2, if for each subset S G of size at most k? 1 the subgraph G n S is connected. We consider the empty graph as connected. Lemma 3.3 Let G be a 2-connected planar graph and a drawing of G. Then the boundary-subgraphs of the faces of G under are cycles. Furthermore, each edge of G is contained in the boundary-subgraph of precisely two faces. We call the boundary-subgraphs of a 2-connected graph G under a drawing the -facial cycles of G. A cycle C in a graph G is chordless, if C is an induced subgraph of G. The cycle C is non-separating if G n C is connected. Lemma 3.4 (Tutte [13]) Let G be a 3-connected planar graph and a drawing of G. Then a cycle of G is -facial if, and only if, it is chordless and non-separating. Note that this Lemma implies Whitney s Theorem that the facial cycles of a 3-connected planar graph do not depend on the drawing and thus, up to homeomorphism, each 3-connected planar graph has a unique drawing in the sphere S 2. A facial cycle of a 3-connected planar graph is a cycle of G that is -facial for some, and hence any, drawing of G The basic cycles In the following, let G be a 3-connected planar graph, a 2 G, and b; c; d distinct neighbors of a (i.e. vertices adjacent to a). We let C G (a; b; c; d) = fc G j C cycle; a; b; c 2 C; d 62 Cg: Intuitively, our idea is that if there is a facial cycle that contains a; b; c, then it will be as far from d as possible, in other words, it will be the innermost cycle in C G (a; b; c; d) (see Figure 2). We show that the relation of being further away from d, or more interior, corresponds to a purely combinatorial partial order relation on w

4 d a b d a b b 1 c c b 2 Figure 3. The cycle C. Figure 2. C G (a; b; c; d) C G (a; b; c; d). If it has precisely one minimal element, then this is the facial cycle that contains a; b; c. In Subsection 3.3, we prove that this minimal cycle can actually be defined in inflationary fixed-point logic. For each C 2 C G (a; b; c; d), we let C = fv 2 G j Each path from v to d intersects Cg; C = hci G, and C G (a; b; c; d) = fc j C 2 C G (a; b; c; d)g: The partial order we are interested in is inclusion on C G (a; b; c; d). Note that by Lemma 3.4 for each facial cycle C 2 C(a; b; c; d) we have C = C. Lemma 3.5 Let C; D 2 C G (a; b; c; d) such that C D. Then C D. The proof is immediate. The following Lemma gives the desired characterization of the facial cycles as the minimal ones. Lemma 3.6 (1) There is a unique C in C G (a; b; c; d) that is minimal with respect to inclusion. (2) If C G (a; b; c; d) contains a facial cycle C then C = C. (3) If C G (a; b; c; d) does not contain a facial cycle then C is not a cycle. Proof. Let be a drawing of G. It induces a cyclic ordering of the neighbors of a. Choosing the right orientation we can arrange the neighbors of a in such a way that b = b 0 ; b 1 ; : : : ; b k = c; : : : ; b l = d; : : : ; b m is an enumeration with respect to this cyclic ordering. For 1 i k let C i be the -facial cycle that contains b i?1 ; a; b i. Let C be the cycle obtained by walking along C 1 from a to b 1, then along C 2 from b 1 to b 2, et cetera, and finally from b k?1 to a along C k (see Figure 3). Note that C 2 C G (a; b; c; d) because if d 2 C i for an i l, then (a; d) would be a chord of the -facial cycle C i, which contradicts Lemma 3.4. P 3 d c a P 2 b w v P 1 Figure 4. C is minimal. To see that C is minimal in C G (a; b; c; d) with respect to inclusion, we prove that C D for all D 2 C G (a; b; c; d). Let D 2 C G (a; b; c; d), v 2 C, and P a path from v to d. We shall prove that P \ D 6= ;. Without loss of generality we can assume that a; b; c 62 P, since they are all elements of D. Let w be the last vertex on P that belongs to C, and let P 3 be the part of P that connects w with d extended by the edge (d; a). Thus P 3 is a path from w to a. Let P 1 be the path on C from w to a through b, and P 2 the path on C from w to a through c (see Figure 4). Then P 1, P 2, P 3 are three internally disjoint paths from w to a. Let F ij, for 1 i < j 3, be defined as in Lemma 3.2. Note that b 2 P 1 n fw; ag and c 2 P 2 n fw; ag. Let P 4 be the path on D from b to c that does not contain a. By the definition of the cycle C, (P 4 ) does not intersect F 12. Thus by Lemma 3.2, P 3 \ P 4 6= ; and thus P \ D 6= ;. This proves the minimality of C. 1 If C G (a; b; c; d) contains a facial cycle, then l = 1 and C is this cycle. Thus (2) follows from the fact that for facial cycles C we have C = C. On the other hand, if C G (a; b; c; d) does not contain a facial cycle, then l > 1 and the edge (a; b 1 ) belongs to the (induced) subgraph C, hence C is not a cycle Defining the minimal and facial cycles Let G, a; b; c; d, and C G (a; b; c; d), C G (a; b; c; d) be as above. 1 There may, however, be another C 2 C G (a; b; c; d) with C = C. P4

5 Let C 2 C G (a; b; c; d) such that C is minimal in C G (a; b; c; d). Let D = G n C. We give an inductive definition of D. Let D 1 = fdg and D i+1 = fv 2 G j9w 2 D i : E G wv; for all i 1. 9C 2 C G (a; b; c; d) : C \ (D i [ fvg) = ;g Proposition 3.7 D = S i1 D i: Proof. An easy induction on i 1 yields the following: Let v 2 D such that there is a path from d to v of length at most (i? 1) that does not intersect C. Then d 2 D i. This implies the forward direction. For the backward direction, let v 2 D i, for some i 1. Then there is a cycle C 2 C G (a; b; c; d) and a path from d to v that does not intersect C (this path goes through S i?1 j=1 D j). By the minimality of C, this path does also not intersect C. Thus v 2 D. 2 Proposition 3.8 There is an IFP-formula '(x; y; z; w; v) such that for all 3-connected planar graphs G and a, b, c, d, e 2 G we have: '(a; b; c; d; e) if, and only if, b, c, d are pairwise distinct neighbors of a and e is contained in the minimal element of C G (a; b; c; d). Definition 3.9 Let G be a 3-connected planar graph. An angle of G is a triple abc 2 G 3, where b; c are distinct neighbors of a and there is a -facial cycle C such that a; b; c 2 C. Note that each angle of a 3-connected planar graph determines a unique facial cycle, for if abc was an angle contained in two faces F 1 and F 2, then fb; cg would separate the graph. The following Lemma is an immediate consequence of the previous proposition. Lemma 3.10 There is an IFP-formula (x; y; z; v) such that for each planar graph G and a; b; c; e 2 G we have: G j= (a; b; c; e) if, and only if, abc is an angle of G and e is contained in the facial cycle determined by abc Defining an order Lemma 3.11 There is an IFP-formula (x; y; z; v; w) such that for any 3-connected planar graph G and angle abc 2 G 3 the binary relation is a linear order of G. fde 2 G 2 j G j= (a; b; c; d; e)g Proof. Let G be a 3-connected planar graph and a 0 b 0 c 0 2 G 3 an angle of G. We inductively define a sequence 1 2 of linear orders with domains D i such that S i1 D i = G, hd i i G is a connected subgraph of G. We define 1 by a 0 1 b 0 1 c 0 : Suppose next that i is defined. CASE 1: There is a facial cycle in D i that is not complete. More precisely, There is a facial cycle C such that C 6 D i, but there is an angle abc 2 D i \ C. We choose the lexicographically minimal angle abc 2 Di 3 (with respect to i ) such that the facial cycle C determined by abc is not completely contained in D i. We let d be the element of C nd i with the smallest neighbor in D i and attach it to the end of i. CASE 2: All facial cycles in D i are complete. We choose the minimal edge ab 2 Di 2 that belongs to a facial cycle C not completely contained in D i. (If there is no such edge, all facial cycles are contained in D i and thus we have D i = G). Without loss of generality we can assume that a i b. There is exactly one c 2 C such that abc is an angle. We attach c to the end of i. Since G is connected, there is an i 1 such that D i = G. By Lemma 3.10, this linear order can be defined in IFP. 2 By Corollary 2.3 this implies Theorem 1.1(1). It also implies a part of Theorem 1.3: Proposition 3.12 The class of 3-connected planar graphs is definable in IFP. Proof. Let (x; y; z; v; w) be the formula obtained by Lemma If a graph G does not contain three vertices a; b; c 2 G such that fde 2 G 2 j G j= (a; b; c; d; e)g is a linear order of G then it is certainly not a 3-connected planar graph. If there are such a; b; c then we have an order available and thus can simulate a P-algorithm that decides whether our graph is 3-connected and planar Definable canonization and block decompositions 4.1. Inflationary fixed-point logic with counting With each finite structure A we associate a two-sorted structure A +. Its restriction to the first sort is A. The second sort consists of the set N A = f0; : : : ; jajg of natural numbers which is equipped with the natural ordering 6.

6 Thus we may write A + = A [ (N A ; 6). We usually refer to the elements of the first sort of A + as points and to the elements of the second sort as numbers. A + = A [ N A denotes the universe of A +. We can now study inflationary fixed-point logic on these two-sorted structures. As a convention, we use symbols x; y; z (and variants) for individual variables ranging over points and ; for individual variables ranging over numbers. If we do not want to specify whether certain individual variables range over points or numbers, we use symbols u; v; w. Relation variables may be two-sorted. We denote the set of all two-sorted IFP-formulas by IFP +. As IFP +, inflationary fixed-point logic with counting IFP+C is a logic that lives in the two-sorted framework described above. The class of IFP+C-formulas is obtained by adding a new term-formation rule to the rules that yield IFP + : If '(x) is a formula and x is a point-variable, then # x '(x) is a number-term. The value of the term # x '(x) in a structure A + is the number of a 2 A such that A + j= '(a), which is a nonnegative integer contained in N A, the number-part of A +. We are mainly interested in those IFP + and IFP+Cformulas that do not have any free number variables. Each such formula defines a relation on the one-sorted structures. When comparing the expressive power of IFP + or IFP+C with that of other logics, we always refer to those formulas that do not have any free number variables Syntactical interpretations We recall the well-known concept of syntactical interpretations. For a formula '(v; w) all of whose free variables occur in the k-tuple v or in the l-tuple w, a structure A, and an l-tuple c 2 A we let '(v; c) A = fb 2 (A + ) k j A j= '(b; c)g: We allow ' to be an IFP+C-formula with free number variables; in this case '(v; c) A is a subset of (A + ) k. Recall that a congruence on a -structure A is an equivalence relation on A such that for all relations R 2, say, of arity r, and for all r-tuples a; b 2 A r with a i b i for all i r we have R A a if, and only if, R A b. We let A= be the factorized -structure. Let k 1 and, = fr 1 ; : : : ; R m g, where R i is r i - ary, be vocabularies. A k-dimensional IFP+C-interpretation of in with parameters w is a sequence (w) = ' uni (v; w); ' = (v 1 ; v 2 ; w); ' R1 (v 1 ; : : : ; v r1 ; w); : : : ; ' Rm (v 1 ; : : : ; v rm ; w) of IFP+C-formulas of vocabulary such that: (i) v and the v i are k-tuples of variables. (ii) The free variables of the formulas of (w) are all among those displayed in the brackets. (iii) For each -structure A and l-tuple c 2 (A + ) l of the appropriate type the binary relation ' = (v 1 ; v 2 ; c) A is a congruence relation on the -structure ' uni (v; c) A ;' R1 (v 1 ; : : : ; v r1 ; c) A ; : : : ; ' Rm (v 1 ; : : : ; v rm ; c) A : (If a relation ' Ri (v 1 ; : : : ; v r1 ; c) A is not contained in (' uni (v; c) A ) ri, we take the intersection.) We let (c) A = ' uni (v; c) A ;' R1 (v 1 ; : : : ; v r1 ; c) A ; : : : ; 4.3. Definable canonization ' Rm (v 1 ; : : : ; v rm ; c) A = '=(v1;v2;c) A: Definition 4.1 We say that a class C of structures admits IFP+C-definable canonization if there is an IFP+Cinterpretation (w) of [ fg in and an IFP+C-formula (w) of vocabulary such that for each structure A 2 C we have: (i) There is a tuple c 2 (A + ) l such that A j= (c). (ii) For all tuples c 2 (A + ) l such that A j= (c) we have (c) A = A, and (c) A is a linear order of (c) A. As a consequence of the ILV-Theorem 2.2 we obtain: Lemma 4.2 Let C be a class of structures admitting IFP+C-definable canonization. Then IFP+C captures P on C. Lemma 4.3 Let C be a class of structures such that there is an IFP+C-formula with parameters that defines an ordering on C. Then C admits IFP+C-definable canonization. Hence in particular the class of 3-connected planar graphs admits IFP+C-definable canonization. A pre-order on a set A is a binary relation that is transitive and connex (that is, for all a; b 2 A we have a b or b a). For each k 1, we reserve a 2k-ary relation symbol k for pre-orders on the k-tuples. For a class C of structures, we let C k = f(a; A k ) j A 2 C; k pre-order on A k g: It is convenient for us to consider trees as directed from the root to the leaves. Nevertheless, when speaking of paths in trees we mean undirected paths. A colored tree is a tree with some additional unary relations, referred to as colors. CT m denotes the class of colored trees with m colors.

7 Lemma 4.4 For each m 0, the class CT 1 m of preordered colored trees admits IFP+C-definable canonization. Proof. To keep things simple, we only consider the case of pure trees, the extension to pre-ordered colored trees is straightforward. We define an IFP+C-formula ' E (; ) such that for all trees T the structure [1; jt j]; ' E (; ) T is isomorphic to T. This clearly yields definable canonization. Let T be a tree. For t 2 T, let S(t) be the subtree of T with root t. We inductively define a ternary relation X of type pointnumbernumber such that for all vertices t 2 T the graph X(t) = [1; js(t )j]; ij 2 [1; js(t )j] 2 Xtij is isomorphic to S(t). For all t, let X t denote the set of all pairs ij such that Xtij. The induction is on the height of the vertex t. If t is a leaf then we just let X t be empty. If t is of height at least one, let t 1 ; : : : ; t l be its children. Suppose that X ti is defined for all i 2 [1; l]. Recall that 6 denotes the natural ordering on the numbers and let X i = (X(t i ); 6). The X i are ordered graphs, and we can order them lexicographically (more precisely, we order their codewords). Let Y 1 < : : : < Y m be a lexicographical enumeration of these ordered graphs. Note that m l. Let n i be the number of occurrences of Y i among the X j (for i m). Now X(t) is the graph obtained by first taking n 1 copies of Y 1, then n 2 copies of the Y 2, et cetera, and finally, after taking n m copies of the Y m, adding another vertex which is then connected to the roots of all the previous trees. Of course once we have defined X(t) we can extract X t. It is straightforward to formalize this definition in the logic IFP+C; we are omitting this here Block decompositions We slightly generalize Robertson s and Seymour s [11] concept of a tree-decomposition of a graph. Definition 4.5 Let D be a class of structures. A block-tree over D is a pair (T; (B t ) t2t ) such that T = (T, E T, P T 1, : : :, P T m ) is a colored tree, B t is either the empty structure or in D for all t 2 T, and for all t; u; v 2 T such that v lies on the path from t to u we have B t \ B u B v. The structures B t (for t 2 T ) are called the blocks and the set S t2t B t the block-domain of (T; (B t ) t2t ). The order of (T; (B t ) t2t ) is defined to be maxfb t \B u j t; u 2 T; t 6= ug. For convenience we always assume that the universe T of the tree is disjoint from the block domain. For each vocabulary, we let B;m = fe, P 1, : : :, P m, Ug [ fr j R 2 g; where E and U are binary, P 1 ; : : : ; P m are unary, and the arity of R is 1 plus the arity of R. We can view a block tree (T; (B t ) t2t ) with m colors over a class of -structures as a B;m -structure B, where (i) B = T [ S t2t B t, (ii) E B = E T and P B i = P T i for i = 1; : : : ; m, (iii) fb 2 B j U B tbg = B t for t 2 T, (iv) R B = ftb j t 2 T; b 2 R Bt g for R 2. In particular, this gives us a natural notion of an isomorphism between block-trees. Definition 4.6 Let C; D classes of structures and k; l; m 0. A block-decomposition of C over D of order k with l parameters and m colors is a pair (P; bt) of mappings where (i) P associates a non-empty subset of A l with each structure A 2 C. (ii) bt associates a block-tree over D of order k with m colors with each pair (A; a), where A 2 C; a 2 P (A). (iii) For all A; B 2 C; a 2 P (A); b 2 P (B) we have (A; a) = (B; b) if, and only if, bt(a; a) = bt(b; b). (iv) There is a polynomial time algorithm that computes a codeword of A from each codeword of bt(a; a) (for all A 2 C; a 2 P (A)). (This algorithm does not have to be isomorphism-invariant.) Of course we may also consider block-decompositions without parameters. It is clear how to modify the definition in this case. Definition 4.7 Let (P; bt) be a block-decomposition of a class C of -structures over a class D of -structures. We say that (P; bt) is IFP+C-definable if there is an IFP+Cformula (w) of vocabulary and an IFP+C-interpretation (w) of B;m in such that for all A 2 C we have: (i) (w) A = P (A). (ii) (c) A = bt(a; c) for all c 2 (w) A. Lemma 4.8 Let C be a class of -structures, D a class of -structures and k 0. Suppose that D k admits IFP+C-definable canonization and that there is an IFP+Cdefinable block-decomposition of C over D of order k. Then C 1 admits IFP+C-definable canonization.

8 Proof. For a given structure A 2 C we can define its blocktree, say, (T; (B t ) t2t ). Possibly, this requires some parameters, which we keep fixed in the following. To define a canonical copy of A we proceed in three steps. In the first step we define canonical ordered copies B t of the blocks B t, using the fact that D k admits IFP+C-definable canonization. In the second step we order the ordered structures fb t j t 2 T g lexicographically and thus obtain a pre-ordering of the vertices of T. By Lemma 4.4, we can define a canonical ordered copy of this preordered colored tree. Thus we obtain an ordered copy of our block-tree. Since on ordered structures IFP+C captures P, we can simulate the P-algorithm to retrieve a A from its block-tree and obtain a canonical ordered copy of A. The most difficult step is the first. The reason is that we cannot just canonize each block B t separately, but have to take intersections between the blocks into account. The intake of a block B t is empty if t is the root of T, and the intersection B t \ B u, where u is the mother of t, otherwise. The outlets of B t are the intakes of the blocks associated with ts children. Note that, by the definition of a block tree, all vertices of a block B t that are also contained in some other block B u are either contained in the intake of B t or in an outlet. The size of all intakes and outlets is bounded by the order k of the block decomposition. To canonize the blocks, we proceed by induction on T. Let t be a leaf of T, and let fb 1 ; : : : ; b l g be the intake of B t. Note that l k. For each possible ordering b i1 < : : : < b il we define a preorder k on B k t with two classes by putting the k- tuple (b i1 ; : : : ; b il ; ; b il : : : ; b il ) in the first class and all other k-tuples in the second class. Then (B t ; k ) 2 D k. We define a canonical ordered copy that we denote by B t (b i1 ; : : : ; b il ; b il ; : : : ; b il ). Let t 2 T and t 1 ; : : : ; t m its children. Let fb 1 ; : : : ; b l g be an outlet of B t, say, the outlet at t 5. With each ordering b i1 < : : : < b il of the outlet we can associate an ordered copy B t5 (b i1 ; : : : ; b il ; : : : ; b il ) of the block B t5. This way we can associate an ordered structure with all k-tuples of the form b i1 ; : : : ; b il ; : : : ; b il that appear in an outlet of B t. Actually, it may happen that we have to associate several ordered structures with a k-tuple since two children of t may correspond to the same outlet of B t. These ordered structures can be ordered lexicographically and give rise to a pre-order 0 k on B k t. Now we proceed as in the case of a leaf: We consider the intake of B t and associate another pre-order 00 k on Bt k with each possible ordering of the intake. We merge the two pre-orders and obtain a k, then we canonize (B t ; k ) and obtain an ordered copy of the block B t for each order of the intake.. Of course we can omit the pre-ordering associated with the intake if the intake is empty. Eventually we have defined an ordered copy of each block for each ordering of its intake. The intake of the root is empty, so we only have one ordered copy of its block. This gives an ordering of each of its outlets, and thus of the intakes of its children. For each of the children we thus have one distinguished ordering of its intake and therefore one distinguished ordered copy of its block. This again gives us an ordering of the outlets, and we can proceed to the children of the next generation, et cetera, until we arrive at the leaves and have associated one distinguished ordered copy of its block with each vertex of the tree Decomposing 3-connected graphs We let G denote the class of all graphs and G k the class of k-connected graphs, for k 1. Moreover, we let P denote the class of all planar graphs and P k the class of k-connected planar graphs, for k The connected components. Formally, we can consider the decomposition of a graph into its connected components as a block decomposition bt 1 (without parameters) of the class of all graphs over G 1. Let G be a graph whose connected components are H 1 ; : : : ; H m. To define bt 1 (G) = (T; (B t ) t2t ) we let T be the tree consisting of m + 1-vertices ft 0 ; t 1 ; : : : ; t m g such that E T t 0 t i for i = 1; : : : ; m. We let B t0 = ; and B ti = H i for i = 1; : : : ; m. It is easy to see that bt 1 is an IFP-definable blockdecomposition of order 0. Thus we have proved the following Lemma: Lemma 5.1 There is an IFP-definable block-decomposition of order 0 of G over G 1. Its restriction to P is an (IFPdefinable) block-decomposition (of order 0) of P over P The 2-connected components. Lemma 5.2 There is an IFP-definable block-decomposition of order 1 of G 1 over G 2. Its restriction to P 1 is a block-decomposition of P 1 over P 2. Proof. The 2-connected components of a graph are its maximal 2-connected subgraphs. We define a block-decomposition (P 2 ; bt 2 ) with one parameter of G 2 over G 1 as follows: For each connected graph G we let P 2 (G) = G. We inductively define the block-trees bt 2 (G; a) = (T; (B t ) t2t ) for connected G and a 2 G.

9 If G is 2-connected, we let T be the tree consisting of a single vertex t 0 whose block is G. If a separates G we let H 1 ; : : : ; H m be the connected components of G n fag and K i = hh i [ fagi G, for i = 1; : : : ; m. Then bt 2 (G; a) is obtained by attaching the block-trees bt 2 (K i ; a) to a new root t 0 whose block is the graph with one vertex a and no edges. If G is not 2-connected, but a does not separate G we let H 0 be the 2-connected component of G that contains a. Let a 1 ; : : : ; a m be the vertices separating H 0 from the rest of G, and for each i let H i be the connected component of h(g n H) [ fa 1 ; : : : ; a m gi G that contains a i. Then bt 2 (G; a) is obtained by attaching the block-trees bt 2 (H i ; a i ) to a new root t 0 whose block is H 0. It is straightforward to show that bt 2 is an IFP-definable block-decomposition The 3-connected components. It is much harder to decompose a graph into its 3- connected components than into its 1 or 2-connected components. Let K denote the class of all cycles. Lemma 5.3 There is an IFP-definable block-decomposition of order 2 of G 2 over G 3 [ K. Its restriction to P 2 is a block-decomposition of P 2 over P 3 [ K. There are different ways to decompose 2-connected graphs, but essentially they are all equivalent. The 3-block decomposition we have chosen is more or less the same as in [14]. A proof of the following two lemmas can be found there. We need some additional notation. Let G be a graph. For an edge e 2 E G we let G? e = (G; E G n feg). For an unordered pair fa; bg of vertices a 6= b 2 G we let G + fa; bg = (G; E G [ ffa; bgg). Let k 0. A k-separation of a graph G is a pair (H; K) of subgraphs of G such that H [ K = G, E H \ E K = ;, jh \ Kj = k, and H n K 6= ;, K n H 6= ;. Lemma 5.4 Let G be a 2-connected graph and e = fa; a 0 g 2 E G such that G n fa; a 0 g is connected, but G 0 = G? e is not 2-connected. Then there is a unique sequence a = a 0 ; a 1 ; : : : ; a m = a 0, where m 2, such that (i) a 1 ; : : : ; a m?1 are precisely the vertices that separate G 0. (ii) For i = 1; : : : ; m? 1 there is a 1-separation (H i ; K i ) of G 0 such that H i \ K i = fa i g and a 0 ; : : : ; a i?1 2 H i, a i+1 ; : : : ; a m+1 2 K i. We let H 0 = K m = G 0. (iii) Let L i = H i \ K i?1, for i = 1; : : : ; m, and L i = (K i ; E Ki [ ffa i?1 ; a i gg). Then L 1 ; : : : ; L m are 2-connected. Furthermore, they are planar if, and only if G is planar. Let e be an edge of a graph G. A 2-separation (H; K) of G is e-minimal if e 2 E H and if for all 2-separations (H 0 ; K 0 ) with e 2 E H0 we have H 0 6 H. Lemma 5.5 Let G be a graph and e 2 E G such that G n fa; a 0 g is connected and G?e is 2-connected. Let (H 1 ; K 1 ), : : :, (H m ; K m ) be a list of all e-minimal 2-separations of G. Furthermore, let H i \ K i = fa i ; a 0 ig (for i = 1; : : : ; m). Let L 0 = T m i=1 H i fa 1 ; a 0 1 ; : : : ; a m; a 0 m g, L 0 = hl 0 i G + fa 1 ; a 0 1 g + + fa m; a 0 m g and L i = K i + fa i ; a 0 ig for 1 i m. Then L 0 is 3-connected, and L 1 ; : : : ; L m are 2-connected is 3-connected. Furthermore, L 0 ; : : : ; L m are planar if, and only if G is planar. Proof (of Lemma 5.3). We define a block decomposition (P 3 ; bt 3 ) with two parameters of G 2 over G 3 [K as follows: For G 2 G 2 we let P 3 (G) = E G. The block trees bt 3 (G; e) are defined inductively: Let G 2 G 2 and e = fa; a 0 g 2 E G. If G is 3-connected, we let bt 3 (G; e) consist of a single vertex t 0 whose block is G. If the set fa; a 0 g separates G, we let H 1 ; : : : ; H m be the connected components of G n fa; a 0 g and K i = hh i [ fa; a 0 gi G, for i = 1; : : : ; m. Then bt 3 (G; e) is obtained by attaching the block-trees bt 3 (K i ; e) to a new root t 0 whose block is (fa; a 0 g; feg). If G n fa; a 0 g is connected, but G? e is not 2- connected, we choose vertices a 0 ; : : : ; a m and graphs L 1 ; : : : ; L m according to Lemma 5.4. For i = 1; : : : ; m we let e i = fa i?1 ; a i g. Then bt 3 (G; e) is obtained by attaching the block-trees bt 3 (L i ; e i ) to a new root t 0. The block of t 0 is the cycle a 0 a 1 : : : a m a 0. If G n fa; a 0 g is connected and G? e is 2-connected, we choose vertices a 1, a 0, 1 : : :, a m, a 0 m and graphs L 0 ; : : : ; K m according to Lemma 5.5. For i = 1; : : : ; m we let e i = fa i ; a 0 ig. By the definition of the graphs L i we have e i 2 E Li. Then bt 3 (G; e) is obtained by attaching the block-trees bt 3 (L i ; e i ), for 1 i m, to a new root t 0. The block of t 0 is L 0. We are not done yet, since there is a problem if we want to reconstruct a graph G from a block-tree bt 3 (G; e) = (T; (B t ) t2t ). With each vertex t 2 T we can associate a distinguished edge e t of B t that has been the e in our induction. Unless t is the root of T, the edge e t can easily be

10 reconstructed from the block tree. Say, e t = fa; a 0 g, and let u 2 T be the mother of t. Then fa; a 0 g = B t \ B u. However, so far the block-tree does not give us any information about whether there is an edge between a and a 0 in our original graph G. To remove this defect, we color all vertices t 2 T for which e t 2 E G in color P 1. Hence our block-decomposition has order 2 and 1 color. It requires some effort, but is nevertheless straightforward to show that (P 3 ; bt 3 ) is an IFP-definable blockdecomposition Putting things together Proof of Theorem 1.1(2). By Lemma 3.11, there is an IFPformula with parameters that defines an ordering on P 3. Hence there is an IFP-formula with parameters that defines an ordering on (P 3 [ K) 2. Thus by Lemma 4.3 the class (P 3 [ K) 2 admits IFP+C-definable canonization. Repeated application of Lemma 4.8 together with the Lemmas of the previous subsections yields that the class P admits IFP+C-definable canonization. Thus by Lemma 4.2, IFP+C captures P on C. 2 Proof of Theorem 1.3. We use the fact that a 2-connected graph is planar if, and only if, the blocks of its decomposition into 3-connected components are (cf. [12]). Since these blocks are definable in IFP and the class of 3-connected planar graphs is definable in IFP, the class of 2-connected planar graphs is definable in IFP. Similarly, a connected graph is planar if, and only if, its 2-connected components are, and an arbitrary graph is planar if, and only if, its connected components are. The respective decompositions are definable in IFP, and thus we obtain the desired result Conclusions The proof of our results falls into two parts. One is graph theoretic and heavily based on particular properties of planar graphs. In the other part we have developed a fairly general machinery to prove capturing results, and I hope that this machinery will have further applications. Although it is known that IFP+C does not capture P on the class of all finite structures [1], I think the results given here can be extended to a much wider class of graphs. The obvious next steps would be classes of graphs of bounded genus and graphs of bounded tree-width. Actually, I hope that IFP+C captures P on all classes of graphs that are closed under minors, except for the class of all graphs. References [1] J. Cai, M. Fürer, and N. Immerman. An optimal lower bound on the number of variables for graph identification. Combinatorica, 12: , [2] R. Diestel. Graph Theory. Springer-Verlag, [3] Y. Gurevich. Toward logic tailored for computational complexity. In M. R. et al., editor, Computation and Proof Theory, volume 1104 of Lecture Notes in Mathematics, pages Springer-Verlag, [4] J. E. Hopcroft and R. Tarjan. Isomorphism of planar graphs (working paper). In R. E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations. Plenum Press, [5] N. Immerman. Upper and lower bounds for first-order expressibility. Journal of Computer and System Sciences, 25:76 98, [6] N. Immerman. Relational queries computable in polynomial time. Information and Control, 68:86 104, [7] N. Immerman. Expressibility as a complexity measure: results and directions. In Proceedings of the 2nd IEEE Symposium on Structure in Complexity Theory, pages , [8] N. Immerman and E. Lander. Describing graphs: A firstorder approach to graph canonization. In A. Selman, editor, Complexity theory retrospective, pages Springer- Verlag, [9] S. Lindell. An analysis of fixed-point queries on binary trees. Theoretical Computer Science, 85:75 95, [10] A. Livchak. The relational model for process control. Automated Documentation and Mathematical Linguistics, 4:27 29, [11] N. Robertson and P. Seymour. Graph minors III. Planar treewidth. Journal of Combinatorial Theory, Series B, pages 49 64, [12] C. Thomassen. Embeddings and minors. In R. Graham, M. Grötschel, and L. Lovász, editors, Handbook of Combinatorics, volume 1, chapter 5, pages Elsevier, [13] W. Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 13: , [14] W. Tutte. Graph Theory. Addison-Wesley, [15] M. Y. Vardi. The complexity of relational query languages. In Proceedings of the 14th ACM Symposium on Theory of Computing, pages , [16] List of open problems in finite model theory. Available at WWW/FMT.html.