General Class Of Graphs Research Articles

Local search: Is brute-force avoidable?

Many local search algorithms are based on searching in the k-exchange neighborhood. This is the set of solutions that can be obtained from the current solution by exchanging at most k elements. As a rule of thumb, the larger k is, the better are the chances of finding an improved solution. However, for inputs of size n, a naïve brute-force search of the k-exchange neighborhood requires nO(k) time, which is not practical even for very small values of k. We show that for several classes of sparse graphs, including planar graphs, graphs of bounded vertex degree and graphs excluding some fixed graph as a minor, an improved solution in the k-exchange neighborhood for many problems can be found much more efficiently. Our algorithms run in time O(τ(k)⋅nc), where τ is a function depending only on k and c is a constant independent of k and n. We demonstrate the applicability of this approach on a variety of problems including r-Center, Vertex Cover, Odd Cycle Transversal, Max-Cut, and Min-Bisection. In particular, on planar graphs, all our algorithms searching for a k-local improvement run in time O(2O(k)⋅n2), which is polynomial for k=O(logn). We complement these fixed-parameter tractable algorithms for k-local search with parameterized intractability results indicating that brute-force search is unavoidable in more general classes of graphs.

On the binary codes with parameters of triply-shortened 1-perfect codes

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the $n$-cube into six cells. An arbitrary binary $(n=2^m-4, 2^{n-m}, 3)$ code $D$, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) from six cells. As a corollary, the codes $C$ and $D$ are completely semiregular; i.e., the weight distribution of such a code depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if $D$ is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable. Keywords: 1-perfect code; triply-shortened 1-perfect code; equitable partition; perfect coloring; weight distribution; distance distribution

Dominating set is fixed parameter tractable in claw-free graphs

We show that the Dominating Set problem parameterized by solution size is fixed-parameter tractable (FPT) in graphs that do not contain the claw ( K 1 , 3 , the complete bipartite graph on four vertices where the two parts have one and three vertices, respectively) as an induced subgraph. We present an algorithm that uses 2 O ( k 2 ) n O ( 1 ) time and polynomial space to decide whether a claw-free graph on n vertices has a dominating set of size at most k . Note that this parameterization of Dominating Set is W [ 2 ] -hard on the set of all graphs, and thus is unlikely to have an FPT algorithm for graphs in general. The most general class of graphs for which an FPT algorithm was previously known for this parameterization of Dominating Set is the class of K i , j -free graphs, which exclude, for some fixed i , j ∈ N , the complete bipartite graph K i , j as a subgraph. For i , j ≥ 2 , the class of claw-free graphs and any class of K i , j -free graphs are not comparable with respect to set inclusion. We thus extend the range of graphs over which this parameterization of Dominating Set is known to be fixed-parameter tractable. We also show that, in some sense, it is the presence of the claw that makes this parameterization of the Dominating Set problem hard. More precisely, we show that for any t ≥ 4 , the Dominating Set problem parameterized by the solution size is W [ 2 ] -hard in graphs that exclude the t -claw K 1 , t as an induced subgraph. Our arguments also imply that the related Connected Dominating Set and Dominating Clique problems are W [ 2 ] -hard in these graph classes. Finally, we show that for any t ∈ N , the Clique problem parameterized by solution size, which is W [ 1 ] -hard on general graphs, is FPT in t -claw-free graphs. Our results add to the small and growing collection of FPT results for graph classes defined by excluded subgraphs, rather than by excluded minors.

Parameterized Complexity of the Spanning Tree Congestion Problem

We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k≥8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k≤3 the problem becomes polynomially time solvable.

On nowhere dense graphs

In this paper, we define and analyze the nowhere dense classes of graphs. This notion is a common generalization of proper minor closed classes, classes of graphs with bounded degree, locally planar graphs, classes with bounded expansion, to name just a few classes which are studied extensively in combinatorial and computer science contexts. In this paper, we show that this concept leads to a classification of general classes of graphs and to the dichotomy between nowhere dense and somewhere dense classes. This classification is surprisingly stable as it can be expressed in terms of the most commonly used basic combinatorial parameters, such as the independence number α , the clique number ω , and the chromatic number χ . The remarkable stability of this notion and its robustness has a number of applications to mathematical logic, complexity of algorithms, and combinatorics. We also express the nowhere dense versus somewhere dense dichotomy in terms of edge densities as a trichotomy theorem.

Equitable edge-colorings of simple graphs

An edge-coloring of a graph G is equitable if, for each v∈V(G), the number of edges colored with any one color incident with v differs from the number of edges colored with any other color incident with v by at most one. A new sufficient condition for equitable edge-colorings of simple graphs is obtained. This result covers the previous results, which are due to Hilton and de Werra, verifies a conjecture made by Hilton recently, and substantially extends it to a more general class of graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:175-197, 2011

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.

QUANTUM NETWORK MODELS AND CLASSICAL LOCALIZATION PROBLEMS

A review is given of quantum network models in class C which, on a suitable 2d lattice, describe the spin quantum Hall plateau transition. On a general class of graphs, however, many observables of such models can be mapped to those of a classical walk in a random environment, thus relating questions of quantum and classical localization. In many cases it is possible to make rigorous statements about the latter through the relation to associated percolation problems, in both two and three dimensions.

The Exact Weighted Independent Set Problem in Perfect Graphs and Related Classes

Abstract The exact weighted independent set (EWIS) problem consists in determining whether a given vertex-weighted graph contains an independent set of given weight. This problem is a generalization of two well-known problems, the NP-complete subset sum problem and the strongly NP-hard maximum weight independent set (MWIS) problem. Since the MWIS problem is polynomially solvable for some special graph classes, it is interesting to determine the complexity of this more general EWIS problem for such graph classes. We focus on the class of perfect graphs, which is one of the most general graph classes where the MWIS problem can be solved in polynomial time. It turns out that for certain subclasses of perfect graphs, the EWIS problem is solvable in pseudo-polynomial time, while on some others it remains strongly NP-complete. In particular, we show that the EWIS problem is strongly NP-complete for bipartite graphs of maximum degree three, but solvable in pseudo-polynomial time for cographs, interval graphs and chordal graphs, as well as for some other related graph classes.

Neural Network for Graphs: A Contextual Constructive Approach

This paper presents a new approach for learning in structured domains (SDs) using a constructive neural network for graphs (NN4G). The new model allows the extension of the input domain for supervised neural networks to a general class of graphs including both acyclic/cyclic, directed/undirected labeled graphs. In particular, the model can realize adaptive contextual transductions, learning the mapping from graphs for both classification and regression tasks. In contrast to previous neural networks for structures that had a recursive dynamics, NN4G is based on a constructive feedforward architecture with state variables that uses neurons with no feedback connections. The neurons are applied to the input graphs by a general traversal process that relaxes the constraints of previous approaches derived by the causality assumption over hierarchical input data. Moreover, the incremental approach eliminates the need to introduce cyclic dependencies in the definition of the system state variables. In the traversal process, the NN4G units exploit (local) contextual information of the graphs vertices. In spite of the simplicity of the approach, we show that, through the compositionality of the contextual information developed by the learning, the model can deal with contextual information that is incrementally extended according to the graphs topology. The effectiveness and the generality of the new approach are investigated by analyzing its theoretical properties and providing experimental results.

Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs

We study graph properties that are testable for bounded-degree graphs in time independent of the input size. Our goal is to distinguish between graphs having a predetermined graph property and graphs that are far from every graph having that property. It is well known that in the bounded-degree graph model (where two graphs are considered “far” if they differ in $\varepsilon n$ edges for a positive constant $\varepsilon$), many graph properties cannot be tested even with a constant or even with a polylogarithmic number of queries. Therefore in this paper we focus our attention on testing graph properties for special classes of graphs. Specifically, we show that every hereditary graph property is testable with a constant number of queries provided that every sufficiently large induced subgraph of the input graph has poor expansion. This result implies that, for example, any hereditary property (e.g., $k$-colorability, $H$-freeness, etc.) is testable in the bounded-degree graph model for planar graphs, graphs with bounded genus, interval graphs, etc. No such results have been known before, and prior to our work, very few graph properties have been known to be testable with a constant number of queries for general graph classes in the bounded-degree graph model.

Coloring parameters for graphs on surfaces

Abstract We overview several coloring parameters with the property that they are bounded on graphs whose (Euler) genus is bounded. Besides the usual chromatic number, we treat the acyclic, the star and the degenerate chromatic number, we discuss some of their relatives, and we determine their dependence on the genus. Extensions to more general classes of graphs (minor-closed families and classes of bounded expansion) are also discussed. The probabilistic method, which is used in proving both upper and lower bounds, gives the dependence of the treated coloring parameters in terms of the maximum degree, a result which is of independent interest.

On finding augmenting graphs

Method of augmenting graphs is a general approach to solve the maximum independent set problem. As the problem is generally NP-hard, no polynomial time algorithms are available to implement the method. However, when restricted to particular classes of graphs, the approach may lead to efficient solutions. A famous example of this type is the maximum matching algorithm: it finds a maximum matching in a graph G , which is equivalent to finding a maximum independent set in the line graph of G . In the particular case of line graphs, the method reduces to finding augmenting (alternating) chains. Recent investigations of more general classes of graphs revealed many more types of augmenting graphs. In the present paper we study the problem of finding augmenting graphs different from chains. To simplify this problem, we introduce the notion of a redundant set. This allows us to reduce the problem to finding some basic augmenting graphs. As a result, we obtain a polynomial time solution to the maximum independent set problem in a class of graphs which extends several previously studied classes including the line graphs.

Ranks of graphs: The size of acyclic orientation cover for deadlock-free packet routing

Given a graph G , the problem is to determine an acyclic orientation of G which minimizes the maximal number of changes of orientation along any shortest path in G . The corresponding value is called the rank of the graph G . The motivation for this graph theoretical problem comes from the design of deadlock-free packet routing protocols [G. Tel, Deadlock-free packet switching networks, in: Introduction to Distributed Algorithms, Cambridge University Press, Cambridge, UK, 1994 (Chapter 5)]. This acyclic orientation covering problem on the shortest path systems has been studied in [J.-C. Bermond, M. Di Ianni, M. Flammini, S. Perennes, Acyclic orientations for deadlock prevention in interconnection networks, in: 23rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG), in: Lecture Notes in Computer Science, vol. 1335, Springer-Verlag, 1997, pp. 52–64] where it was shown that the general problem of determining the rank is NP-complete and some upper and lower bounds on the rank were presented for particular topologies, such as grids, tori and hypercubes. The main unresolved problem stated in [J.-C. Bermond, M. Di Ianni, M. Flammini, S. Perennes, Acyclic orientations for deadlock prevention in interconnection networks, in: 23rd International Workshop on Graph-Theoretic Concepts in Computer Science (WG), in: Lecture Notes in Computer Science, vol. 1335, Springer-Verlag, 1997, pp. 52–64] was to determine the rank values for other well-known interconnection networks and also for more general classes of graphs. In this paper we give a general lower bound argument for the rank problem and apply it to the class of involution-generated Cayley graphs which among others include hypercubes, star graphs, pancake graphs and transposition-tree based graphs [S.B. Akers, B. Krishnamurthy, A group-theoretic model for symmetric interconnection networks, IEEE Transactions on Computers 38 (4) (1989) 555–565]. We also present a large class L C P ( T , SP ) of graphs with constant rank. This class of graphs is defined as the layered cross product [S. Even, A. Litman, Layered cross product—A technique to construct interconnection networks, Networks 29 (1997) 219–223] of layered trees and series–parallel graphs and includes among others butterflies, Beneš networks, fat-trees and meshes of trees. For some special topologies, improved lower bounds on the rank are also presented. We consider some of the modified versions of the rank problem as well.

CYCLIC PROPERTIES OF TRIANGULAR GRID GRAPHS

It is known that all 2-connected, linearly convex triangular grid graphs, with only one exception, are hamiltonian (Reay and Zamfirescu, 2000). In the paper, it is shown that this result holds for a wider class of connected, locally connected triangular grid graphs and, with more exceptions, even for some general class of graphs. It is also shown that the HAMILTONIAN CYCLE problem is NP-complete for triangular grid graphs.

Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover

Our goal in this paper is the development of fast algorithms for recognizing general classes of graphs. We seek algorithms whose complexity can be expressed as a linear function of the graph size p...

Incremental maintenance of shortest distance and transitive closure in first-order logic and SQL

Given a database, the view maintenance problem is concerned with the efficient computation of the new contents of a given view when updates to the database happen. We consider the view maintenance problem for the situation when the database contains a weighted graph and the view is either the transitive closure or the answer to the all-pairs shortest-distance problem ( APSD ). We give incremental algorithms for APSD , which support both edge insertions and deletions. For transitive closure, the algorithm is applicable to a more general class of graphs than those previously explored. Our algorithms use first-order queries, along with addition (+) and less-than (<) operations ( FO (+,<)); they store O ( n 2 ) number of tuples, where n is the number of vertices, and have AC 0 data complexity for integer weights. Since FO (+,<) is a sublanguage of SQL and is supported by almost all current database systems, our maintenance algorithms are more appropriate for database applications than nondatabase query types of maintenance algorithms.

Fast fixed-parameter tractable algorithms for nontrivial generalizations of vertex cover

Our goal in this paper is the development of fast algorithms for recognizing general classes of graphs. We seek algorithms whose complexity can be expressed as a linear function of the graph size plus an exponential function of k, a natural parameter describing the class. In particular, we consider the class W k ( G ) , where for each graph G in W k ( G ) , the removal of a set of at most k vertices from G results in a graph in the base graph class G . (If G is the class of edgeless graphs, W k ( G ) is the class of graphs with bounded vertex cover.) When G is a minor-closed class such that each graph in G has bounded maximum degree, and all obstructions of G (minor-minimal graphs outside G ) are connected, we obtain an O ( ( g + k ) | V ( G ) | + ( fk ) k ) recognition algorithm for W k ( G ) , where g and f are constants (modest and quantified) depending on the class G . If G is the class of graphs with maximum degree bounded by D (not closed under minors), we can still obtain a running time of O ( | V ( G ) | ( D + k ) + k ( D + k ) k + 3 ) for recognition of graphs in W k ( G ) . Our results are obtained by considering bounded-degree minor-closed classes for which all obstructions are connected graphs, and showing that the size of any obstruction for W k ( G ) is O ( tk 7 + t 7 k 2 ) , where t is a bound on the size of obstructions for G . A trivial corollary of this result is an upper bound of ( k + 1 ) ( k + 2 ) on the number of vertices in any obstruction of the class of graphs with vertex cover of size at most k. These results are of independent graph-theoretic interest.

Near-shortest and K-shortest simple paths

We present a new algorithm for enumerating all near-shortest simple (loopless) s-t paths in a graph G = (V, E) with nonnegative edge lengths. Letting n = |V| and m = |E|, the time per path enumerated is O(nS(n, m)) given a user-selected shortest-path subroutine with complexity O(S(n, m)). When coupled with binary search, this algorithm solves the corresponding K-shortest paths problem (KSPR) in O(KnS(n, m)(log n+ log cmax)) time, where cmax is the largest edge length. This time complexity is inferior to some other algorithms, but the space complexity is the best available at O(m). Both algorithms are easy to describe, to implement and to extend to more general classes of graphs. In computational tests on grid and road networks, our best polynomial-time algorithm for KSPR appears to be at least an order of magnitude faster than the best algorithm from the literature. However, we devise a simpler algorithm, with exponential worst-case complexity, that is several orders of magnitude faster yet on those test problems. A minor variant on this algorithm also solves “KSPU,” which is analogous to KSPR but with loops allowed. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(2), 98–109 2005

On the independent dominating set polytope

In this paper, we consider the independent dominating set polytope. We give a complete linear description of that polytope when the graph is reduced to a cycle. This description uses a general class of valid inequalities introduced in [T.M. Contenza, Some results on the dominating set polytope, Ph.D. Dissertation, University of Kentucky, 2000]. We devise a polynomial time separation algorithm for these inequalities. As a consequence, we obtain a polynomial time cutting plane algorithm for the minimum (maximum) independent dominating set problem on a cycle. We also introduce a lifting operation called twin operation, and discuss some polyhedral consequences. In particular, we show that the above results can be extended to a more general class of graphs.