A Simple Algorithm for the Graph Minor Decomposition { Logic meets Structural Graph Theory{ Martin Grohe∗ Ken-ichi Kawarabayashiyz Bruce Reedx Abstract which says that in any infinite collection of finite graphs A key result of Robertson and Seymour's graph minor there is one that is a minor of another. As with other theory is a structure theorem stating that all graphs deep results in mathematics, the body of theory devel- excluding some fixed graph as a minor have a tree oped for the proof of the graph minor theorem has also decomposition into pieces that are almost embeddable found applications elsewhere, both within graph theory in a fixed surface. Most algorithmic applications of and computer science. Most of these applications rely graph minor theory rely on an algorithmic version of this not only on the general techniques developed by Robert- result. However, the known algorithms for computing son and Seymour to handle graph minors, but also on such graph minor decompositions heavily rely on the one particular structural result (proved in [29]), which is very long and complicated proofs of the existence of such central to the whole theory. It describes the structure of decompositions, essentially they retrace these proofs all graphs G which do not contain some fixed graph H and show that all steps are algorithmic. as a minor. At a high level, the theorem says that ev- In this paper, we give a simple quadratic time algo- ery such a graph can be decomposed into a collection rithm for computing graph minor decompositions. The of graphs each of which can be \nearly" embedded into best previously known algorithm due to Kawarabayashi a bounded-genus surface; the pieces can be assembled and Wollan runs in cubic time and is far more compli- in a tree structure to obtain the original graph. In the cated. following, we refer to such a decomposition as a graph Our algorithm combines techniques from logic and minor decomposition. structural graph theory, or more precisely, a variant of Starting with Robertson and Seymour's cubic time Courcelle's Theorem stating that monadic second-order algorithm for the disjoint path problem [28], a substan- logic formulas can be evaluated in linear time on graphs tial body of work on \algorithmic graph minor theory" of bounded tree width and Robertson and Seymour's so emerged (e.g. [7, 8, 9, 14, 15, 17]). Most of the re- called Weak Structure Theorem. sults give efficient (exact or approximation) algorithms for various hard problems on classes of graphs with ex- 1 Introduction cluded minors, but some go beyond such classes [28, 15]. Almost all of these results rely, either directly or indi- A graph H is a minor of a graph G if H can be obtained rectly through other results, on the existence of graph from a subgraph of G by contracting edges. The theory minor decompositions (i.e., Robertson and Seymour's of graph minors was developed by Robertson and Sey- structure theorem) and on efficient algorithms for com- mour in a series of 23 papers published over more than puting these decompositions. twenty-five years. The aim of that series of papers was Several such algorithms are known. The third au- the proof of a single result: the graph minor theorem, thor of this paper was maybe the first to point out that Robertson-Seymour's original proof of the structure the- ∗RWTH Aachen University, Germany. Email address: [email protected] orem (which requires almost 400 pages) gives rise to a yNational Institute of Informatics and JST ERATO polynomial time algorithm to construct the decompo- Kawarabayashi Project, Tokyo, Japan. sitions. Demaine, Hajiaghayi, and Kawarabayashi [9] Email address: k [email protected] give a lengthy proof for constructing it, which builds zResearch partly supported by Japan Society for the Promo- tion of Science, Grant-in-Aid for Scientific Research, , by C & on many structural graph minor results. The run- k C Foundation, by Kayamori Foundation and by Inoue Research ning time of this algorithm is O(n ) for a k that de- Award for Young Scientists. pends on the size of the excluded minor. Dawar, xCanada Research Chair in Graph Theory, McGill University, Grohe, and Kreutzer [6] give an fixed-parameter algo- Montreal Canada and Project Mascotte, INRIA, Laboratoire I3S, rithm that, however, computes a \weaker" decompo- CNRS, Sophia-Antipolis, France. sition into pieces that have bounded local tree width Email address: [email protected] after removing a bounded number of vertices. Recently, graph minor decomposition of the bounded-tree-width Kawarabayashi and Wollan [18] found a dramatically graph G0 in quadratic time. Then, in the third step, our shorter proof for the graph minor decomposition theo- algorithm re-inserts the vertices deleted in the first step rem (cutting off around 300 pages of the original graph and extends the decomposition from G0 to G. minor papers), which yields a cubic time algorithm for The paper is organised as follows: After general pre- computing graph minor decompositions. However, all liminaries in Section 2, we formally define graph minor these algorithms are deeply entrenched in rather heavy decompositions in Section 3 and state the structure the- structural graph theory. Essentially, they are algorith- orem. We prove claim (A) from above in Section 4 and mic proofs of the structure theorem. claim (B) in Section 5. We put everything together in In this paper, we give a simple quadratic time al- Section 6. gorithm to construct a graph minor decomposition. We take a completely different approach than the previous 2 Preliminaries algorithms. We take the existence of a decomposition For all integers m; n, we denote the set m; m + for granted and just try to find one. We reduce struc- 1; : : : ; n , which is empty if m > n, by [m; n],f and we tural graph theory to a minimum, but combine it with let [n] :=g [1; n]. We use a standard graph theoretic ter- tools from logic. The correctness proof of our algorithm minology and notation. The set of all neighbours of a 1 essentially fits within this conference paper, which is vertex w or a set W of vertices in a graph G is denoted quite remarkable when compared with previous algo- by N G(w) and N G(W ), respectively, and for a subgraph rithms. H of G we let N G(H) = N G(V (H)). Our main technical contributions are the following. A tree decomposition of a graph G is a pair (T;Y ), (A) We prove one graph theoretic result, which roughly where T is a tree and Y is a family Yt t V (T ) of vertex sets Y V (G), such that thef followingj 2 twog says that if there is a vertex v deep inside a grid t ⊆ in a graph G, then no matter how we obtain a properties hold: graph minor decomposition of G v, we can put − (1) t V (T ) Yt = V (G), and every edge of G has both v back into this decomposition and thus obtain a 2 ends in some Yt. decomposition of G. S (2) If t; t0; t00 V (T ) and t0 lies on the path in T from (B) We prove that \near embeddings" (in the sense 2 t to t00, then Y Y 00 Y 0 . required for graph minor decompositions) of graphs t \ t ⊆ t in a bounded genus surface are definable in monadic For every node t V (T ), we call Y the bag at t. The 2 t second logic. torso at t is the graph Ht obtained from the induced Using these two results, our algorithm proceeds as subgraph G[Yt] by adding edges between all vertices v; w such that v; w Y Y for some neighbour u N T (t). follows. Repeatedly applying (A), it deletes vertices 2 t \ u 2 from the input graph G until it arrives at a graph G0 It is sometimes convenient to view the tree in a tree that no longer has a large grid. By the Excluded Grid decomposition as rooted, and will freely do so. Theorem [25], G0 has bounded tree width. Applying The adhesion of a tree decomposition (T;Y ) is max Y Y tu E(T ) if E(T ) = , and 0 if an extension of Courcelle's well known theorem [2, 4], fj t \ uj j 2 g 6 ; stating that monadic second-order formulas can be E(T ) = . The width of (T;Y ) is max Yt t V (T ) 1,; and the tree width tw(G) of G isfj definedj j as2 evaluated in linear time, to the formulas obtained in g − (B), we obtain a linear time algorithm for constructing the minimum width taken over all tree decompositions near embeddings of bounded tree width graphs. We of G. By a well-known result due to Bodlaender [3], can even construct such near embeddings \locally" there is an algorithm that, given a graph G and an with respect to a tangle. (A tangle [27] may be integer w, decides if the tree width of G is at most w and viewed as a structure describing a highly connected computes a tree decomposition of G of width at most w O(w3) region in a graph, and a near embedding is local if it is. The running time of the algorithm is 2 n, where n = G . with respect to the tangle if the part of the graph j j properly embedded falls within this region.) Now we can use a (fairly simple and by now standard) generic 3 Structure Theorems construction going back to [27] for computing a \global" 3.1 Tangles Let G be a graph.