Automated Testing and Interactive Construction of Unavoidable Sets for Graph Classes of Small Path-width Oliver Bachtler and Irene Heinrich Abstract We present an interactive framework that, given a membership test for a graph class G and a number k ∈ N, finds and tests unavoidable sets for the class Gk of graphs in G of path-width at most k. We put special emphasis on the case that G is the class of cubic graphs and tailor the algorithm to this case. In particular, we introduce the new concept of high-degree-first path-decompositions, which yields highly efficient prun- ing techniques. Using this framework we determine all extremal girth values of cubic graphs of path-width k for all k ∈ {3,..., 10}. Moreover, we determine all smallest graphs which take on these extremal girth values. As a further application of our framework we characterise the extremal cubic graphs of path-width 3 and girth 4. 1 Introduction A set of graphsis unavoidable for a graph class if every graph in the class contains an isomorphic copy of a graph in the set. Unavoidable sets are extensively used in ⊲ interactive proofs, for example Appel and Haken’s proof of the famous 4-colouring conjecture, cf. [1, 2], ⊲ structural graph theory, where unavoidable sets give insights into the be- haviour of the considered class and are, hence, of intrinsic interest, see for example [8], ⊲ recursive algorithms and inductive proofs, for example [4, 13]. arXiv:2010.08373v1 [math.CO] 16 Oct 2020 While discharging [10] is a tool to find unavoidable structures for colouring problems and Ramsey theory [9] studies unavoidable sets in extremal graph theory, there is no generic approach for finding or checking unavoidable sets. Frequently, tedious case distinctions are necessary to prove that some set is indeed unavoidable for a considered class, cf. [4, 13]. Our contribution. We present a recipe for the automatic construction and testing of unavoidable sets for classes of small path-width. Let a class G with a membership test and a number k ∈ N be given. Our interactive framework finds and tests unavoidable sets for all classes of the form Gk := {G ∈ G : G is of path-width at most k}. 1 To this end, we describe an algorithm which investigates the hypothesis that U is unavoidable for Gk. After each phase i, the algorithm either ⊲ returns a counterexample of order k + i, or ⊲ returns NONE, guaranteeing that there is no counterexample to the hypoth- esis, or ⊲ guarantees that there is no counterexample of order k + i and proceeds with phase i + 1. This framework can easily be adapted to a check for unavoidable induced sub- graphs or for unavoidable minors. We use the algorithm to determine bounds on the girth of cubic graphs of small path-width. Recall that the girth of a graph is the minimum length amongst its cycles. Consider the following example, in which the algorithm is used as a black box: Let G be the class of cubic graphs and write Ui for the set {C3,...,Ci}, where Cℓ denotes a cycle on ℓ vertices. To investigate on the maximal girth values of cubic graphs depending on their path-width, we set ξ : N≥3 → N, k 7→ max{g : there is a simple cubic graph of girth g and path-width k}. We want to precisely determine ξ(k) for small values of k. For the case k = 3 we first check whether all cubic graphs of path-width 3 have a triangle by running the algorithm on the set U3 and k = 3. It returns 3 the K3,3 which has girth 4. Hence, U3 is not unavoidable for G . We extend the set U3 to U4. The algorithm now returns NONE, meaning that this set is, indeed, unavoidable and ξ(3) = 4. With this method, we can find the value of ξ(k) for k =3,..., 7, as shown in Table 1. k 3 3 4 5 5 6 6 7 U U3 U4 U4 U4 U5 U5 U6 U6 Result K3,3 NONE NONE Petersen NONE Heawood NONE NONE Table 1: Algorithmic results for cubic graphs of path-width k and unavoidable structures U. This describes how the algorithm is to be used in general: start with a set of structures, potentially the empty set, and run the algorithm on it. If the set is not unavoidable, the provided counterexample can be used to extend the set of structures, either by adding the graph itself or a subgraph. Repeat this process until the set of structures is unavoidable. Subsequently to obtaining the results in Table 1 with the help of a computer, we managed to prove the theorems below by hand, making use of the formalisms introduced to prove the correctness of the algorithm: Theorem 1.1. For all k ∈ N≥3 the following inequality is satisfied: 2 10 ξ(k) ≤ 3 k + 3 . Furthermore we prove the upper bounds on the girth in the theorem below, which are an improvement on the previous bound for k ≤ 13. The equalities obtained for k ≤ 10 use the examples listed in the table of Theorem 1.3, whose girth values coincide with the determined upper bounds. 2 Theorem 1.2. The values of ξ for small values of k are shown in the table below: k 3 4 5 6 7 8 9 10 ξ(k) 4 4 5 6 6 7 8 8 Additionally, ξ(k) ≤ k − 2 holds for all k ≥ 10. We give a complete list of the minimal graphs of path-width k and girth ξ(k) for all k ∈{3,..., 10}. Theorem 1.3. For k ∈ {3,..., 10}\{4} there is a unique smallest graph of path-width k and girth ξ(k). There are two smallest graphs of path-width 4 and girth 4. The results are summarised in the following table: k smallest cubic graphs of path-width k and girth ξ(k) 3 K3,3 4 cube, twisted cube 5 Petersen graph 6 Heawood graph 7 Pappus graph 8 McGee graph 9 Tutte Coxeter graph 10 G(10), where G(10) denotes the unique cubic graph of path-width 10 and girth 8. We refer to Figure 1 for drawings of all graphs in the above list. Recall that a (d, g)-cage is a minimal d-regular graph of girth g. The study of cages dates back to [15]. A recent survey on this topic is [12]. If d = 3, then there is a unique (3,g)-cage for all g ∈{3,..., 8}. Clearly, if a cubic cage of girth ξ(k) has path-width k, then it appears in the table. Thus, it is not surprising that several of the graphs above are cages: the K3,3, the Petersen graph, the Heawood graph and the Tutte Coxeter graph. As a further application of our algorithm we obtain that {K3,3, G1,...,G6} (see Figure 3) is unavoidable for the class of cubic graphs of path-width 3 and girth 4. We exploit this to prove the following classification. Theorem 1.4. (i) Any 3-connected cubic graph of path-width 3 and girth 4 can be obtained from a K3,3 by a finite number of the construction steps C1 and C2 (see Figure 3). (ii) Any cubic graph of path-width 3 and girth 4 can be constructed from a K3,3 by applying a finite number of the construction steps C1,...,C6 (see Figure 3). Techniques for the algorithm. In essence, the algorithm checks graphs G ∈ Gk by simulating the traversal of a smooth path-decomposition (see [3] or Section 2 of this paper). The algorithm runs in phases and manages a queue. At the beginning of phase i, the queue contains all pairs of the form (Vi, Gi) which satisfy 3 =∼ Figure 1: All graphs of path-width k and girth ξ(k) for k ∈{3, 4, 5, 6, 7, 8, 9, 10}. First row: K3,3, cube, twisted cube, Petersen graph, Heawood graph. Middle row: Pappus graph, McGee graph, Tutte Coxeter graph. Last row: Two draw- ings of the unique graph of girth 8 and path-width 10, which we denote by G(10). k ⊲ Gi is a subgraph of some graph G ∈ G which has a smooth path- decomposition with Vi as its ith bag. In particular, |V (Gi)| = k + i, ⊲ Gi contains all information provided by the bags preceding Vi in the path- decomposition, ⊲ Gi is a potential subgraph of a counterexample to U being unavoidable for Gk. The algorithm checks for the current pair (Vi, Gi) whether adding additional k edges to Gi results in a graph in G that avoids all graphs in U. If this is the case, then the obtained graph is returned as a certificate that U is not k unavoidable for G . Otherwise, (Vi, Gi) is replaced by new pairs (Vi+1, Gi+1), where Gi+1 is a supergraph of Gi of order |V (Gi)| + 1. If the queue is empty, then the algorithm guarantees that U is unavoidable for Gk. 4 To drastically limit the amount of additional pairs created, we heavily rely on isomorphism rejection to prune the resulting search tree. In order to avoid a combinatorial explosion, we refine smooth path-decompositions to high-degree- first path-decompositions. These are invaluable when tailoring the algorithm to cubic graphs since they alleviate the need to branch out into new pairs with multiple different candidates for bags in a step where the associated graph has a vertex of degree at least 2. Instead they give us a unique bag for all subsequent pairs.