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Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs

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Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs Structure Theorem and Isomorphism Test for Graphs with Excluded Topological Subgraphs Martin Grohe Dániel Marx∗ Institut für Informatik Computer and Automation Research Institute, Humboldt-Universität zu Berlin Hungarian Academy of Sciences (MTA SZTAKI) Berlin, Germany Budapest, Hungary [email protected] [email protected] ABSTRACT Keywords We generalize the structure theorem of Robertson and Sey- topological minors, fixed-parameter tractability, graph iso- mour for graphs excluding a fixed graph H as a minor to morphism graphs excluding H as a topological subgraph. We prove that for a fixed H, every graph excluding H as a topological subgraph has a tree decomposition where each part is either 1. INTRODUCTION \almost embeddable" to a fixed surface or has bounded de- We say that a graph H is a minor of G if H can be ob- gree with the exception of a bounded number of vertices. tained from G by deleting vertices, deleting edges, and con- Furthermore, such a decomposition is computable by an al- tracting edges. A graph G is H-minor free if H is not a minor gorithm that is fixed-parameter tractable with parameter of G. Robertson and Seymour [25] proved a structure theo- SHS. rem for the class of H-minor graphs: roughly speaking, every We present two algorithmic applications of our structure H-minor free graph can be decomposed in a way such that theorem. To illustrate the mechanics of a \typical" appli- each part is \almost embeddable" into a fixed surface. This cation of the structure theorem, we show that on graphs structure theorem has important algorithmic consequences: excluding H as a topological subgraph, Partial Dominat- many natural computational problems become easier when ing Set (find k vertices whose closed neighborhood has restricted to H-minor free graphs [4, 13, 6, 15, 14, 5, 10]. maximum size) can be solved in time f(H; k) ⋅ nO(1) time. These algorithmic results can be thought of as far-reaching More significantly, we show that on graphs excluding H as generalizations of algorithms on planar graphs and bounded- a topological subgraph, Graph Isomorphism can be solved genus surfaces. in time nf(H). This result unifies and generalizes two pre- A more general way of defining restricted classes of graphs viously known important polynomial-time solvable cases of is to exclude topological subgraphs instead of minors. A Graph Isomorphism: bounded-degree graphs [18] and H- graph H is a topological subgraph (or topological minor) of minor free graphs [22]. The proof of this result needs a graph G if a subdivision of H is a subgraph of G. It is easy generalization of our structure theorem to the context of to see that if H is a topological subgraph of G, then H is invariant treelike decomposition. also a minor of G. Thus the class of graphs excluding H as a topological subgraph is a more general class than H-minor free graphs. Categories and Subject Descriptors One can ask if graphs excluding H as a topological sub- graph admit a similar structure theorem as H-minor free F.2 [Theory of Computing]: Analysis of Algorithms and graphs. However, graphs excluding a topological subgraph Problem Complexity; G.2.2 [Mathematics of Comput- can be much more general. For example, no 3-regular graph ing]: Discrete Mathematics|Graph Theory can contain a subdivision of K5 (as K5 is 4-regular). There- fore, the class of graphs excluding K5 as a topological sub- General Terms graph includes in particular every 3-regular graph. This suggests that it is unlikely that this class can be also char- Algorithms acterized by (almost) embeddability into surfaces. Nevertheless, our first result is a structure theorem for ∗Research supported by the European Research Council graphs excluding a graph H as a topological subgraph. We (ERC) grant 280152. prove that, in some sense, only the bounded-degree graphs make this class more general than H-minor free graphs. More precisely, we prove a structure theorem that decom- poses graphs excluding H as a topological subgraph into ′ Permission to make digital or hard copies of all or part of this work for almost bounded-degree parts and into H -minor free parts ′ ′ personal or classroom use is granted without fee provided that copies are (for some other graph H ). The H -minor free parts can not made or distributed for profit or commercial advantage and that copies be further refined into almost-embeddable parts using the bear this notice and the full citation on the first page. To copy otherwise, to structure theorem of Robertson and Seymour [25], to obtain republish, to post on servers or to redistribute to lists, requires prior specific our main structural result (see Corollary 4.4 for the precise permission and/or a fee. statement): STOC’12, May 19–22, 2012, New York, New York, USA. Copyright 2012 ACM 978-1-4503-1245-5/12/05 ...$10.00. Theorem 1.1 (informal). For every fixed graph H, it contains several tree decompositions. The Invariant De- every graph excluding H as a topological subgraph has a tree composition Theorem (Section 8) generalizes the structure decomposition where every torso theorem by giving an algorithm that computes a treelike de- composition in a way that the decompositions obtained for (i) either has bounded degree with the exception of a bounded isomorphic graphs are isomorphic. Then the Lifting Lemma number of vertices, or (Section 9) formalizes the bottom-up strategy informally de- (ii) almost embeddable into a surface of bounded genus. scribed in the previous paragraph: if we can compute treelike Furthermore, such a decomposition can be computed in time decompositions for a class of graphs in an invariant way and f(H) ⋅ SV (G)SO(1) for some computable function f. we have a canonization algorithm for the bags, then we have a canonization algorithm for this class of graphs. Although Our structure theorem allows us to lift problems that are the idea is simple, in order to encode the child bags, we have tractable on both bounded-degree graphs and on H-minor to state this algorithmic result in a more general form: in- free graphs to the class of graphs excluding H as a topolog- stead of graphs, we have to work with weighted relational ical subgraph. We demonstrate this principle on the Par- structures. This makes the statement and proof of the Lift- tial Dominating Set problem (find k vertices whose closed ing Lemma more technical. Putting together these results, neighborhood is maximum). Following a bottom-up dy- we obtain: namic programming approach, we solve the problem in each bag of the tree decomposition (using the fact that the prob- Theorem 1.3. For every fixed graph H, Graph Isomor- lem can be solved in linear-time on both bounded-degree phism can be solved in polynomial-time restricted to graphs and on almost-embeddable graphs). excluding H as a topological subgraph. Theorem 1.2. Partial Dominating Set can be solved Let us quickly remark that it is unlikely that Theorem 1.3 in time f(k; H) ⋅ nO(1) when restricted to graphs excluding could be generalized to all classes of graphs with bounded H as a topological subgraph. expansion, as the isomorphism problem on such a class can be as hard as on general graphs. To see this, consider two One could prove similar results for other basic problems such ′ graphs G1 and G2 on n-vertices and let us obtain G1 and as Independent Set or Dominating Set. However, a re- ′ ′ G2 by subdividing each edge with n new vertices. Now G1 sult of Dvorak et al. [7] shows that problems expressible in ′ and G2 have bounded expansion and they are isomorphic if first-order logic can be solved in linear time on classes of and only if G1 and G2 are. graphs having bounded expansion, and therefore on graphs Actually, we not only obtain a polynomial time isomor- excluding H as a topological subgraph. The problems In- phism test, but also a polynomial time canonisation algo- dependent Set and Dominating Set (for a fixed k) can rithm. Our theorem generalizes and unifies the results of be expressed in first-order logic, thus the analogs of Theo- Babai and Luks [18, 2] on bounded-degree graphs and of rem 1.2 for these problems follow from [7]. On the other Ponomarenko [22] on H-minor free graphs. Let us remark hand, Partial Dominating Set is not expressible in first- that Ponomarenko's result implies that there is a polynomial order logic, hence the techniques of Dvorak et al. [7] do not time isomorphism test for all classes of graphs of bounded apply to this problem. genus, which has been proved earlier by Filotti and Mayer [8] The main algorithmic result of the paper concerns the and Miller [21], and for all classes of graphs of bounded tree Graph Isomorphism problem (given graphs G1 and G2, width, which was also proved later (independently) by Bod- decide if they are isomorphic). Graph Isomorphism is laender [3]. Miller [20] gave a common generalization of the known to be polynomial-time solvable for bounded-degree bounded degree and bounded genus classes to classes that graphs [18, 2] and for H-minor free graphs [22, 9]. In fact, he called k-contractible. These classes do not seem to have for these classes of graphs, even the more general canoniza- a simple graph-theoretic characterization; they are defined tion problem can be solved in polynomial time: there is an in terms of properties of the automorphism groups needed algorithm labeling the vertices of the graph with positive for the algorithm.
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