Graph Minors Research Articles

Split Contraction

The edit operation that contracts edges, which is a fundamental operation in the theory of graph minors, has recently gained substantial scientific attention from the viewpoint of Parameterized Complexity. In this article, we examine an important family of graphs, namely, the family of split graphs, which in the context of edge contractions is proven to be significantly less obedient than one might expect. Formally, given a graph G and an integer k , S PLIT C ONTRACTION asks whether there exists X ⊆ E ( G ) such that G / X is a split graph and | X | ≤ k . Here, G / X is the graph obtained from G by contracting edges in X . Guo and Cai [Theoretical Computer Science, 2015] claimed that S PLIT C ONTRACTION is fixed-parameter tractable. However, our findings are different. We show that S PLIT C ONTRACTION , despite its deceptive simplicity, is W[1]-hard. Our main result establishes the following conditional lower bound: Under the Exponential Time Hypothesis, S PLIT C ONTRACTION cannot be solved in time 2 o (ℓ 2 )⋅ n O (1), where ℓ is the vertex cover number of the input graph. We also verify that this lower bound is essentially tight. To the best of our knowledge, this is the first tight lower bound of the form 2 o (ℓ 2 )⋅ n O (1) for problems parameterized by the vertex cover number of the input graph. In particular, our approach to obtain this lower bound borrows the notion of harmonious coloring from Graph Theory, and might be of independent interest.

Contractible edges in 3-connected graphs that preserve a minor

Let G be a 3-connected graph with a 3-connected (or sufficiently small) simple minor H. We establish that G has a forest F with at least ⌈(|G|−|H|+1)/2⌉ edges such that G/e is 3-connected with an H-minor for each e∈E(F). Moreover, we may pick F with |G|−|H| edges provided G is triangle-free. These results are sharp. Our result generalizes a previous one by Ando et al., which establishes that a 3-connected graph G has at least ⌈|G|/2⌉ contractible edges. As another consequence, each triangle-free 3-connected graph has an spanning tree of contractible edges. Our results follow from a more general theorem on graph minors, a splitter theorem, which is also established here.

Excluded minors in cubic graphs

Let G be a cubic graph, with girth at least five, such that for every partition X,Y of its vertex set with |X|,|Y|≥7 there are at least six edges between X and Y. We prove that if there is no homeomorphic embedding of the Petersen graph in G, and G is not one particular 20-vertex graph, then either•G∖v is planar for some vertex v; or•G can be drawn with crossings in the plane, but with only two crossings, both on the infinite region. We also prove several other theorems of the same kind.

Rooted Complete Minors in Line Graphs with a Kempe Coloring

It has been conjectured that if a finite graph has a vertex coloring such that the union of any two color classes induces a connected graph, then for every set $T$ of vertices containing exactly one member from each color class there exists a complete minor such that $T$ contains exactly one member from each branching set. Here we prove the statement for line graphs.

Graph minors and the linear reducibility of Feynman diagrams

We look at a graph property called reducibility which is closely related to a condition developed by Brown to evaluate Feynman integrals. We show for graphs with a fixed number of external momenta, that reducibility with respect to both Symanzik polynomials is graph minor closed. We also survey the known forbidden minors and the known structural results. This gives some structural information on those Feynman diagrams which are reducible.

Induced and weak induced arboricities

We define the induced arboricity of a graph G, denoted by ia(G), as the smallest k such that the edges of G can be covered with k induced forests in G.For a class F of graphs and a graph parameter p, let p(F)=sup{p(G)∣G∈F}. We show that ia(F) is bounded from above by an absolute constant depending only on F, that is ia(F)≠∞ if and only if χ(F∇12)≠∞, where F∇12 is the class of 12-shallow minors of graphs from F andχ is the chromatic number.As a main contribution of this paper, we provide bounds on ia(F) when F is the class of planar graphs, the class of d-degenerate graphs, or the class of graphs having tree-width at most d. Specifically, we show that if F is the class of planar graphs, then 8≤ia(F)≤10.In addition, we establish similar results for so-called weak induced arboricities and star arboricities of classes of graphs.

Near-linear time constant-factor approximation algorithm for branch-decomposition of planar graphs

We give an algorithm that for an input planar graph G of n vertices and an integer k, in min{O(nlog3n),O(nk2)} time either constructs a branch-decomposition of G of width at most (2+δ)k, where δ>0 is a constant, or a (k+1)×k+12 cylinder minor of G implying bw(G)>k, where bw(G) is the branchwidth of G. This is the first Õ(n) time constant-factor approximation for branchwidth/treewidth and largest grid/cylinder minors of planar graphs and improves the previous min{O(n1+ϵ),O(nk2)} (where ϵ>0 is a constant) time constant-factor approximations. For a planar graph G andk=bw(G), a branch-decomposition of width at most (2+δ)k and a g×g2 cylinder/grid minor with g=kβ, where β>2 is a constant, can be computed by our algorithm in min{O(nlog3nlogk),O(nk2logk)} time.

On the Hyperbolicity Constant in Graph Minors

A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims in this work is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph $$\,G/e$$ obtained from the (simple or non-simple) graph G by contracting an arbitrary edge e from it. We prove that H is hyperbolic if and only if G is hyperbolic, for many minors H of G, even if H is obtained from G by contracting and/or deleting infinitely many edges.

Explicit bounds for graph minors

Let Σ be a surface with boundary bd(Σ), L be a collection of k disjoint bd(Σ)-paths in Σ, and P be a non-separating bd(Σ)-path in Σ. We prove that there is a homeomorphism ϕ:Σ→Σ that fixes each point of bd(Σ) and such that ϕ(L) meets P at most 2k times.With this theorem, we derive explicit constants in the graph minor algorithms of Robertson and Seymour (1995) [10]. We reprove a result concerning redundant vertices for graphs on surfaces, but with explicit bounds. That is, we prove that there exists a computable integer t:=t(Σ,k) such that if v is a ‘t-protected’ vertex in a surface Σ, then v is redundant with respect to any k-linkage.

A note on immersion minors and planarity

Graph minors play an important role in graph theory. The focus of this paper is on immersion minors and their relationship to planarity. In general, planar graphs can have non-planar immersion minors. This paper shows that by placing a simple restriction on the immersion-minor operations, all immersion minors of a planar graph are planar. This then allows one to easily obtain a characterization of planar graphs using immersion minors. A dual form of this characterization, as well as an extension to binary matroids, are also considered.

Linear Time Parameterized Algorithms for S ubset F eedback V ertex S et

In the S ubset F eedback V ertex S et (S ubset FVS) problem, the input is a graph G on n vertices and m edges, a subset of vertices T , referred to as terminals, and an integer k . The objective is to determine whether there exists a set of at most k vertices intersecting every cycle that contains a terminal. The study of parameterized algorithms for this generalization of the F eedback V ertex S et problem has received significant attention over the past few years. In fact, the parameterized complexity of this problem was open until 2011, when two groups independently showed that the problem is fixed parameter tractable. Using tools from graph minors,, Kawarabayashi and Kobayashi obtained an algorithm for S ubset FVS running in time O ( f ( k )ċ n 2 m ) [SODA 2012, JCTB 2012]. Independently, Cygan et al. [ICALP 2011, SIDMA 2013] designed an algorithm for S ubset FVS running in time 2 O ( k log k ) ċ n O (1) . More recently, Wahlström obtained the first single exponential time algorithm for S ubset FVS, running in time 4 k ċ n O (1) [SODA 2014]. While the 2 O ( k ) dependence on the parameter k is optimal under the Exponential Time Hypothesis, the dependence of this algorithm as well as those preceding it, on the input size is at least quadratic. In this article, we design the first linear time parameterized algorithms for S ubset FVS. More precisely, we obtain the following new algorithms for S ubset FVS. — A randomized algorithm for S ubset FVS running in time O (25.6 k ċ ( n + m )). — A deterministic algorithm for S ubset FVS running in time 2 O ( k log k ) ċ ( n + m ). Since it is known that assuming the Exponential Time Hypothesis, S ubset FVS cannot have an algorithm running in time 2 o ( k ) n O (1) , our first algorithm obtains the best possible asymptotic dependence on both the parameter as well as the input size. Both of our algorithms are based on “cut centrality,” in the sense that solution vertices are likely to show up in minimum size cuts between vertices sampled from carefully chosen distributions.

Finding and Using Expanders in Locally Sparse Graphs

We show that every locally sparse graph contains a linearly sized expanding subgraph. For constants c_1>c_2>1, 0<\alpha<1, a graph G on n vertices is called a (\c_1,c_2,\alpha)-graph if it has at least c_1n edges, but every vertex subset W\subset V(G) of size |W|łe \alpha n spans less than c_2|W| edges. We prove that every (c_1,c_2,\alpha)-graph with bounded degrees contains an induced expander on linearly many vertices. The proof can be made algorithmic. We then discuss several applications of our main result to random graphs, to problems about embedding graph minors, and to positional games.

K6 minors in large 6-connected graphs

Jørgensen conjectured that every 6-connected graph with no K6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs.

K6 minors in 6-connected graphs of bounded tree-width

We prove that every sufficiently large 6-connected graph of bounded tree-width either has a K6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently large 6-connected graphs. Jørgensen conjectured that it holds for all 6-connected graphs.

Topological Minors of Cover Graphs and Dimension

AbstractWe show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result that is supporting a conjectured generalization of the previous result. We prove that ‐free posets whose cover graphs exclude a fixed graph as a topological minor contain only standard examples of size bounded in terms of k.

Tight Lower Bounds on Graph Embedding Problems

We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph G to graph H cannot be done in time | V ( H )| o (| V ( G )|) . We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of | V ( H )| o (| V ( H )|) -time algorithm deciding if graph G is a subgraph of H . For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems. Moreover, as a consequence of our reductions, conditional lower bounds follow for other related problems such as Locally Injective Homomorphism, Graph Minors, Topological Graph Minors, Minimum Distortion Embedding and Quadratic Assignment Problem.

Minors in graphs of large [formula omitted]-girth

For every r∈N, let θr denote the graph with two vertices and r parallel edges. The θr-girth of a graph G is the minimum number of edges of a subgraph of G that can be contracted to θr. This notion generalizes the usual concept of girth which corresponds to the case r=2. In Kühn and Osthus (2003), Kühn and Osthus showed that graphs of sufficiently large minimum degree contain clique-minors whose order is an exponential function of their girth. We extend this result for the case of θr-girth and we show that the minimum degree can be replaced by some connectivity measurement. As an application of our results, we prove that, for every fixed r, graphs excluding as a minor the disjoint union of kθr’s have treewidth O(k⋅logk).

Formula omitted]-minors in optimal [formula omitted]-planar graphs

We discuss the existence of minors of given graphs in optimal 1-planar graphs. As our first main result, we prove that for any graph H, there exists an optimal 1-planar graph which contains H as a topological minor. Next, we consider minors of complete graphs. It is easily obtained from Mader’s result (Mader, 1968) that every optimal 1-planar graph has a K6-minor. In the paper, we characterize optimal 1-planar graphs having no K7-minor.

Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets

Using quantum annealing to solve an optimization problem requires minor embedding a logic graph into a known hardware graph. In an effort to reduce the complexity of the minor embedding problem, we introduce the minor set cover (MSC) of a known graph $${\mathcal {G}}$$G: a subset of graph minors which contain any remaining minor of the graph as a subgraph. Any graph that can be embedded into $${\mathcal {G}}$$G will be embeddable into a member of the MSC. Focusing on embedding into the hardware graph of commercially available quantum annealers, we establish the MSC for a particular known virtual hardware, which is a complete bipartite graph. We show that the complete bipartite graph $$K_{N,N}$$KN,N has a MSC of N minors, from which $$K_{N+1}$$KN+1 is identified as the largest clique minor of $$K_{N,N}$$KN,N. The case of determining the largest clique minor of hardware with faults is briefly discussed but remains an open question.

Polynomial Bounds for the Grid-Minor Theorem

One of the key results in Robertson and Seymour’s seminal work on graph minors is the grid-minor theorem (also called the excluded grid theorem ). The theorem states that for every grid H , every graph whose treewidth is large enough relative to | V ( H )| contains H as a minor. This theorem has found many applications in graph theory and algorithms. Let f ( k ) denote the largest value such that every graph of treewidth k contains a grid minor of size ( f ( k ) × f ( k )). The best previous quantitative bound, due to recent work of Kawarabayashi and Kobayashi, and Leaf and Seymour, shows that f ( k )=Ω(√log k /log log k ). In contrast, the best known upper bound implies that f ( k ) = O (√ k /log k ). In this article, we obtain the first polynomial relationship between treewidth and grid minor size by showing that f ( k ) = Ω( k δ ) for some fixed constant δ &gt; 0, and describe a randomized algorithm, whose running time is polynomial in | V ( G )| and k , that with high probability finds a model of such a grid minor in G .