Apex Graph Research Articles

K-apices of minor-closed graph classes. I. Bounding the obstructions

Let G be a minor-closed graph class. We say that a graph G is a k-apex of G if G contains a set S of at most k vertices such that G∖S belongs to G. We denote by Ak(G) the set of all graphs that are k-apices of G. We prove that every graph in the obstruction set of Ak(G), i.e., the minor-minimal set of graphs not belonging to Ak(G), has order at most 2222poly(k), where poly is a polynomial function whose degree depends on the order of the minor-obstructions of G. This bound drops to 22poly(k) when G excludes some apex graph as a minor.

Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering

We prove the following theorem. Given a planar graph $G$ and an integer $k$, it is possible in polynomial time to randomly sample a subset $A$ of vertices of $G$ with the following properties: (i) $A$ induces a subgraph of $G$ of treewidth $\mathcal{O}(\sqrt{k}\log k)$, and (ii) for every connected subgraph $H$ of $G$ on at most $k$ vertices, the probability that $A$ covers the whole vertex set of $H$ is at least $(2^{\mathcal{O}(\sqrt{k}\log^2 k)}\cdot n^{\mathcal{O}(1)})^{-1}$, where $n$ is the number of vertices of $G$. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound $2^{\mathcal{O}(\sqrt{k} \log^2 k)} n^{\mathcal{O}(1)}$. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph, examples of such problems include Directed $k$-Path, Weighted $k$-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface.

K -apices of Minor-closed Graph Classes. II. Parameterized Algorithms

Let 𝒢 be a minor-closed graph class. We say that a graph G is a k -apex of 𝒢 if G contains a set S of at most k vertices such that G\S belongs to 𝒢 . We denote by 𝒜 k ( 𝒢 ) the set of all graphs that are k -apices of 𝒢 . In the first paper of this series, we obtained upper bounds on the size of the graphs in the minor-obstruction set of 𝒜 k ( 𝒢 ), i.e., the minor-minimal set of graphs not belonging to 𝒜 k ( 𝒢 ). In this article, we provide an algorithm that, given a graph G on n vertices, runs in time 2 poly (k) ⋅ n 3 and either returns a set S certifying that G ∈ 𝒜 k ( 𝒢 ), or reports that G ∉ 𝒜 k ( 𝒢 ). Here poly is a polynomial function whose degree depends on the maximum size of a minor-obstruction of 𝒢 . In the special case where 𝒢 excludes some apex graph as a minor, we give an alternative algorithm running in 2 poly (k) ⋅ n 2 -time.

On Certain Aspects of Topological Indices

A topological index, also known as connectivity index, is a molecular structure descriptor calculated from a molecular graph of a chemical compound which characterizes its topology. Various topological indices are categorized based on their degree, distance, and spectrum. In this study, we calculated and analyzed the degree-based topological indices such as first general Zagreb index M r G , geometric arithmetic index GA G , harmonic index H G , general version of harmonic index H r G , sum connectivity index λ G , general sum connectivity index λ r G , forgotten topological index F G , and many more for the Robertson apex graph. Additionally, we calculated the newly developed topological indices such as the AG 2 G and Sanskruti index for the Robertson apex graph G.

All Minor-Minimal Apex Obstructions with Connectivity Two

A graph is an apex graph if it contains a vertex whose deletion leaves a planar graph. The family of apex graphs is minor-closed and so it is characterized by a finite list of minor-minimal non-members. The long-standing problem of determining this finite list of apex obstructions remains open. This paper determines the $133$ minor-minimal, non-apex graphs that have connectivity two.

Planar Graphs Have Bounded Queue-Number

We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath et al. [66] from 1992. The key to the proof is a new structural tool called layered partitions , and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number. Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. [31] that graphs in a proper minor-closed class have low treewidth colourings.

Layered separators in minor-closed graph classes with applications

Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω(n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed classes. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded size with respect to a different measure, called the width. We prove, for example, that planar graphs and graphs of bounded Euler genus admit layered separators of bounded width. More generally, we characterise the minor-closed classes that admit layered separators of bounded width as those that exclude a fixed apex graph as a minor.We use layered separators to prove O(log⁡n) bounds for a number of problems where O(n) was a long-standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results with a O(log⁡n) bound on the nonrepetitive chromatic number of graphs excluding a fixed topological minor, and a logO(1)⁡n bound on the queue-number of graphs excluding a fixed minor. Only for planar graphs were logO(1)⁡n bounds previously known. Our results imply that every n-vertex graph excluding a fixed minor has a 3-dimensional grid drawing with nlogO(1)⁡n volume, whereas the previous best bound was O(n3/2).

To Approximate Treewidth, Use Treelength!

Tree-likeness parameters have proven their utility in the design of efficient algorithms on graphs. In this paper, we relate the structural tree-likeness of graphs with their metric tree-likeness. To this end, we establish new upper bounds on the diameter of minimal separators in graphs. We prove that in any graph $G$, the diameter of any minimal separator $S$ in $G$ is at most $\lfloor \ell(G) / 2\rfloor \cdot (|S|-1)$, with $\ell(G)$ the length of a longest isometric cycle in $G$. Our result relies on algebraic methods and on the cycle basis of graphs. We improve our bound for the graphs admitting a distance preserving elimination ordering, for which we prove that any minimal separator $S$ has diameter at most $2 \cdot (|S|-1)$. We use our results to prove that the treelength $tl(G)$ of any graph $G$ is at most $\lfloor \ell(G) / {2}\rfloor$ times its treewidth $tw(G)$. In addition, we prove that, for any graph $G$ that excludes an apex graph $H$ as a minor, $tw(G) \leq c_H \cdot tl(G)$ for some constan...

Beyond bidimensionality: Parameterized subexponential algorithms on directed graphs

In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, k-Leaf Out-Branching, which is to find an oriented spanning tree with at least k leaves, we obtain an algorithm solving the problem in time 2O(klogk)n+nO(1) on directed graphs whose underlying undirected graph excludes some fixed graph H as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2O(k)n+nO(1). The second example is a generalization of the Directed Hamiltonian Path problem, namely k-Internal Out-Branching, which is to find an oriented spanning tree with at least k internal vertices. We obtain an algorithm solving the problem in time 2O(klogk)+nO(1) on directed graphs whose underlying undirected graph excludes some fixed apex graph H as a minor. Finally, we observe that on these classes of graphs, the k-Directed Path problem is solvable in time O((1+ε)knf(ε)), for any ε>0, where f is some function of ε.Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem-specific combinatorial structures, and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs.

Graphs Without Large Apples and the Maximum Weight Independent Set Problem

An appleAk is the graph obtained from a chordless cycle Ck of length k ≥ 4 by adding a vertex that has exactly one neighbor on the cycle. The class of apple-free graphs is a common generalization of claw-free graphs and chordal graphs, two classes enjoying many attractive properties, including polynomial-time solvability of the maximum weight independent set problem. Recently, Brandstadt et al. showed that this property extends to the class of apple-free graphs. In the present paper, we study further generalization of this class called graphs without large apples: these are (Ak, Ak+1, . . .)-free graphs for values of k strictly greater than 4. The complexity of the maximum weight independent set problem is unknown even for k = 5. By exploring the structure of graphs without large apples, we discover a sufficient condition for claw-freeness of such graphs. We show that the condition is satisfied by bounded-degree and apex-minor-free graphs of sufficiently large tree-width. This implies an efficient solution to the maximum weight independent set problem for those graphs without large apples, which either have bounded vertex degree or exclude a fixed apex graph as a minor.

Subdivisions in apex graphs

The Kelmans-Seymour conjecture states that every 5-connected nonplanar graph contains a subdivided K 5. Certain questions of Mader propose a “plan” towards a possible resolution of this conjecture. One part of this plan is to show that every 5-connected nonplanar graph containing $K^{-}_{4}$ or K 2,3 as a subgraph has a subdivided K 5. Recently, Ma and Yu showed that every 5-connected nonplanar graph containing $K^{-}_{4}$ as a subgraph has a subdivided K 5. We take interest in K 2,3 and prove that every 5-connected nonplanar apex graph containing K 2,3 as a subgraph contains a subdivided K 5. The result of Ma and Yu can be used in a short discharging argument to prove that every 5-connected nonplanar apex graph contains a subdivided K 5; here we propose a longer proof whose merit is that it avoids the use of discharging and employs a more structural approach; consequently it is more amenable to generalization.

Vertex insertion approximates the crossing number of apex graphs

An apex graph is a graph G from which only one vertex v has to be removed to make it planar. We show that the crossing number of such G can be approximated up to a factor of Δ ( G − v ) ⋅ d ( v ) / 2 by solving the vertex inserting problem, i.e. inserting a vertex plus incident edges into an optimally chosen planar embedding of a planar graph. Since the latter problem can be solved in polynomial time, this establishes the first polynomial fixed-factor approximation algorithm for the crossing number problem of apex graphs with bounded degree. Furthermore, we extend this result by showing that the optimal solution for inserting multiple edges or vertices into a planar graph also approximates the crossing number of the resulting graph.

Parameterizing cut sets in a graph by the number of their components

For a connected graph G=(V,E), a subset U⊆V is a disconnected cut if U disconnects G and the subgraph G[U] induced by U is disconnected as well. A cut U is a k-cut if G[U] contains exactly k(≥1) components. More specifically, a k-cut U is a (k,ℓ)-cut if V∖U induces a subgraph with exactly ℓ(≥2) components. The Disconnected Cut problem is to test whether a graph has a disconnected cut and is known to be NP-complete. The problems k-Cut and (k,ℓ)-Cut are to test whether a graph has a k-cut or (k,ℓ)-cut, respectively. By pinpointing a close relationship to graph contractibility problems we show that (k,ℓ)-Cut is in P for k=1 and any fixed constant ℓ≥2, while it is NP-complete for any fixed pair k,ℓ≥2. We then prove that k-Cut is in P for k=1 and NP-complete for any fixed k≥2. On the other hand, for every fixed integer g≥0, we present an FPT algorithm that solves (k,ℓ)-Cut on graphs of Euler genus at most g when parameterized by k+ℓ. By modifying this algorithm we can also show that k-Cut is in FPT for this graph class when parameterized by k. Finally, we show that Disconnected Cut is solvable in polynomial time for minor-closed classes of graphs excluding some apex graph.

Spanners in sparse graphs

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. If S is required to be a tree then S is called a tree t-spanner of G. In 1998, Fekete and Kremer showed that on unweighted planar graphs deciding whether G admits a tree t-spanner is polynomial time solvable for t ⩽ 3 and is NP-complete when t is part of the input. They also left as an open problem if the problem is polynomial time solvable for every fixed t ⩾ 4 . In this work we resolve the open question of Fekete and Kremer by proving much more general results: • The problem of finding a t-spanner of treewidth at most k in a given planar graph G is fixed parameter tractable parameterized by k and t. Moreover, for every fixed t and k, the running time of our algorithm is linear. • Our technique allows to extend the result from planar graphs to much more general classes of graphs. An apex graph is a graph that can be made planar by the removal of a single vertex. We prove that the problem of finding a t-spanner of treewidth k is fixed parameter tractable on graphs that do not contain some fixed apex graph as a minor, i.e. on apex-minor-free graphs. The class of apex-minor-free graphs contains planar graphs and graphs of bounded genus. • Finally, we show that the tractability border of the t-spanner problem cannot be extended beyond the class of apex-minor-free graphs and in this sense our results are tight. In particular, for every t ⩾ 4 , the problem of finding a tree t-spanner is NP-complete on K 6 -minor-free graphs.

Cliques, minors and apex graphs

In this paper, we proved the following result: Let G be a ( k + 2 ) -connected, non- ( k − 3 ) -apex graph where k ≥ 2 . If G contains three k -cliques, say L 1 , L 2 , L 3 , such that | L i ∩ L j | ≤ k − 2 ( 1 ≤ i < j ≤ 3 ) , then G contains a K k + 2 as a minor. Note that a graph G is t - apex if G − X is planar for some subset X ⊆ V ( G ) of order at most t . This theorem generalizes some earlier results by Robertson, Seymour and Thomas [N. Robertson, P.D. Seymour, R. Thomas, Hadwiger conjecture for K 6 -free graphs, Combinatorica 13 (1993) 279–361.], Kawarabayashi and Toft [K. Kawarabayashi, B. Toft, Any 7-chromatic graph has K 7 or K 4 , 4 as a minor, Combinatorica 25 (2005) 327–353] and Kawarabayashi, Luo, Niu and Zhang [K. Kawarabayashi, R. Luo, J. Niu, C.-Q. Zhang, On structure of k -connected graphs without K k -minor, Europ. J. Combinatorics 26 (2005) 293–308].

Separation Results for Separated Apex NLC and NCE Graph Languages

Relations among various types of node replacement graph languages are known in the literature but there has remained a gap in the hierarchy of these graph languages. The present paper fills this gap by proving a separation result for k-separated apex graph languages and by extending a known result for k-separated graph languages. This yields, together with other known relations, a complete hierarchy of NLC, NCE, eNCE graph languages and their k-separated, linear, and/or apex subclasses.

Face Covers and the Genus Problem for Apex Graphs

A graph G is an apex graph if it contains a vertex w such that G−w is a planar graph. It is easy to see that the genus g(G) of the apex graph G is bounded above by τ−1, where τ is the minimum face cover of the neighbors of w, taken over all planar embeddings of G−w. The main result of this paper is the linear lower bound g(G)⩾τ/160 (if G−w is 3-connected and τ>1). It is also proved that the minimum face cover problem is NP-hard for planar triangulations and that the minimum vertex cover is NP-hard for 2-connected cubic planar graphs. Finally, it is shown that computing the genus of apex graphs is NP-hard.

Diameter and Treewidth in Minor-Closed Graph Families

It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to bounded-genus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph families.

Apex graphs with embeddings of face-width three

An apex graph is a graph which has a vertex whose removal makes the resulting graph planar. Embeddings of apex graphs having face-width three are characterized. Surprisingly, there are such embeddings of arbitrarily large genus. This solves a problem of Robertson and Vitray. We also give an elementary proof of a result of Robertson, Seymour, and Thomas (1993) that any embedding of an apex graph in a nonorientable surface has face-width at most two.

Context-free graph languages of bounded degree are generated by apex graph grammars

The apex graph grammars generate precisely the context-free graph languages of bounded degree, independently of whether one considers hyperedge replacement systems or (boundary or confluent) NLC or edNCE graph grammars. The main feature of apex graph grammars is that nodes cannot be "passed" from nonterminal to nonterminal. The proof is based on a normal form result for arbitrary hyperedge replacement systems that forbids "passing chains". This generalizes Greibach Normal Form.