Important Classes Of Graphs Research Articles

The Cover Time of Deterministic Random Walks

The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how quickly this "deterministic random walk" covers all vertices (or all edges). We present general techniques to derive upper bounds for the vertex and edge cover time and derive matching lower bounds for several important graph classes. Depending on the topology, the deterministic random walk can be asymptotically faster, slower or equally fast as the classic random walk. We also examine the short term behavior of deterministic random walks, that is, the time to visit a fixed small number of vertices or edges.

Numbering action vertices in workflow graphs

Numbering action vertices in workflow graphsWorkflow graphs, consisting of actions, events, and logical switches, are used to model business processes. In order to easily identify the actions within a workflow graph, it is useful to number them in such a way that the numbering reflects the structure of the workflow. However, available tools offer only rudimental numbering schemes. In the paper, a set of natural requirements is defined that a logical numbering should fulfill. It is investigated under what conditions there is an appropriate numbering at all, when it is uniquely defined by the set of requirements, and when it can be computed efficiently. It is shown that for an important special class of workflow graphs, namely, structured workflow graphs, the answer to all these questions is affirmative. For general workflow graphs, a set of requirements is presented that can always be fulfilled, but the numbering is not necessarily unique. An algorithm based on a depth-first search can be used to compute an appropriate numbering efficiently.

On diameter and inverse degree of a graph

The inverse degree r ( G ) of a finite graph G = ( V , E ) is defined as r ( G ) = ∑ v ∈ V 1 deg v , where deg v is the degree of vertex v . We establish inequalities concerning the sum of the diameter and the inverse degree of a graph which for the most part are tight. We also find upper bounds on the diameter of a graph in terms of its inverse degree for several important classes of graphs. For these classes, our results improve bounds by Erdős et al. (1988) [5], and by Dankelmann et al. (2008) [4].

Cayley graphs as classifiers for data mining: The influence of asymmetries

The endomorphism monoids of graphs have been actively investigated. They are convenient tools expressing asymmetries of the graphs. One of the most important classes of graphs considered in this framework is that of Cayley graphs. Our paper proposes a new method of using Cayley graphs for classification of data. We give a survey of recent results devoted to the Cayley graphs also involving their endomorphism monoids.

On a cycle partition problem

Let G be any graph and let c ( G ) denote the circumference of G . We conjecture that for every pair c 1 , c 2 of positive integers satisfying c 1 + c 2 = c ( G ) , the vertex set of G admits a partition into two sets V 1 and V 2 , such that V i induces a graph of circumference at most c i , i = 1 , 2 . We establish various results in support of the conjecture; e.g. it is observed that planar graphs, claw-free graphs, certain important classes of perfect graphs, and graphs without too many intersecting long cycles, satisfy the conjecture. This work is inspired by a well-known, long-standing, analogous conjecture involving paths.

Algorithmic Graph Minor Theory: Improved Grid Minor Bounds and Wagner’s Contraction

We explore three important avenues of research in algorithmic graph-minor theory, which all stem from a key min-max relation between the treewidth of a graph and its largest grid minor. This min-max relation is a keystone of the Graph Minor Theory of Robertson and Seymour, which ultimately proves Wagner’s Conjecture about the structure of minor-closed graph properties. First, we obtain the only known polynomial min-max relation for graphs that do not exclude any fixed minor, namely, map graphs and power graphs. Second, we obtain explicit (and improved) bounds on the min-max relation for an important class of graphs excluding a minor, namely, K 3,k -minor-free graphs, using new techniques that do not rely on Graph Minor Theory. These two avenues lead to faster fixed-parameter algorithms for two families of graph problems, called minor-bidimensional and contraction-bidimensional parameters, which include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal parameters, dominating set, edge dominating set, R-dominating set, connected dominating set, connected edge dominating set, connected R-dominating set, and unweighted TSP tour. Third, we disprove a variation of Wagner’s Conjecture for the case of graph contractions in general graphs, and in a sense characterize which graphs satisfy the variation. This result demonstrates the limitations of a general theory of algorithms for the family of contraction-closed problems (which includes, for example, the celebrated dominating-set problem). If this conjecture had been true, we would have had an extremely powerful tool for proving the existence of efficient algorithms for any contraction-closed problem, like we do for minor-closed problems via Graph Minor Theory.

A New Algorithm for Finding Minimal Cycle-Breaking Sets of Turns in a Graph

We consider the problem of constructing a minimal cycle-breaking set of turns for a given undirected graph. This problem is important for deadlock-free wormhole routing in computer and communication networks, such as Networks of Workstations. The proposed Cycle Breaking algorithm, or CB algorithm, guarantees that the constructed set of prohibited turns is minimal and that the fraction of the prohibited turns does not exceed 1=3 for any graph. The computational complexity of the proposed algorithm is O(N 2 ¢), where N is the number of vertices, and ¢ is the maximum node degree. The memory complexity of the algorithm is O(N¢). We provide lower bounds on the minimum size of cycle-breaking sets for connected graphs. Further, we construct minimal cycle-breaking sets and establish bounds on the minimum fraction of prohibited turns for two important classes of graphs, namely, t-partite graphs and graphs with small degrees. The upper bounds are tight and demonstrate the optimality of the CB algorithm for certain classes of graphs. Results of computer simulations illustrate the superiority of the proposed CB algorithm as compared to the well-known and the widely used Up/Down technique.

On strongly circular-perfectness

Abstract Perfect graphs constitute a well-studied graph class with a rich structure. Circular-perfect graphs as introduced by Zhu are a natural superclass of perfect graphs defined by means of a more general coloring concept and form an important class of χ-bound graphs with the smallest non-trivial χ-binding function χ ( G ) ⩽ ω ( G ) + 1 . Apart from perfect graphs, circular-perfect graphs include all convex-round graphs and outerplanar graphs. A linear description of facets of stable set polytopes of circular-perfect graphs is still unknown, though convex-round graphs and outerplanar graphs are rank-perfect. In this paper, we exhibit an infinite class of non rank-perfect circular-perfect graphs. We introduce strongly circular-perfectness: a circular-perfect graph is said to be strongly circular-perfect if its complement is also circular-perfect. This subclass of circular-perfect graphs includes perfect graphs, odd holes and antiholes. We show that there are infinitely many non rank-perfect strongly circular-perfect graphs and fully characterize triangle-free minimal strongly circular-imperfect graphs: it turns out that these graphs are very close to odd holes.

On two equimatchable graph classes

A graph G is said to be equimatchable if every matching in G extends to (i.e., is a subset of) a maximum matching. In this paper, we use the Gallai–Edmonds decomposition theory for matchings to determine the equimatchable members of two important graph classes. We find that there are precisely 23 3-connected planar graphs (i.e., 3-polytopes) which are equimatchable and that there are only two cubic equimatchable graphs.

Branching Random Walks on Quasi-Transitive Graphs

The branching random walk on a regular graph turns out to be particularly easy to analyse using results for the corresponding simple random walk. In this way, one can show that there is an intermediate phase of weak survival if and only if the graph is nonamenable. No such simple analysis holds more generally, and it is known that the nonamenability equivalence does not extend to general connected graphs of bounded degree (although we observe that it does hold for such graphs if the branching random walk is modified in a certain natural way). The most important general class of (bounded degree, connected) graphs for which it is thought that the equivalence may hold is that of quasi-transitive graphs: we show that this is indeed the case.

On External-Memory Planar Depth First Search

Even though a large number of I/O-efficient graph algorithms have been developed, a number of fundamental problems still remain open. For example, no space- and I/O-efficient algorithms are known for depth-first search or breadth-first search in sparse graphs. In this paper we present two new results on I/O-efficient depth-first search in an important class of sparse graphs, namely undirected embedded planar graphs. We develop a new efficient depth-first search algorithm and show how planar depth-first search in general can be reduced to planar breadth-first search. As part of the first result we develop the first I/O-efficient algorithm for finding a simple cycle separator of a biconnected planar graph. Together with other recent reducibility results, the second result provides further evidence that external memory breadth-first search is among the hardest problems on planar graphs.

On zero-error source coding with decoder side information

Let (X,Y) be a pair of random variables distributed over a finite product set V/spl times/W according to a probability distribution P(x,y). The following source coding problem is considered: the encoder knows X, while the decoder knows Y and wants to learn X without error. The minimum zero-error asymptotic rate of transmission is shown to be the complementary graph entropy of an associated graph. Thus, previous results in the literature provide upper and lower bounds for this minimum rate (further, these bounds are tight for the important class of perfect graphs). The algorithmic aspects of instantaneous code design are considered next. It is shown that optimal code design is NP-hard. An optimal code design algorithm is derived. Polynomial-time suboptimal algorithms are also presented, and their average and worst case performance guarantees are established.

Towards area requirements for drawing hierarchically planar graphs

Hierarchical graphs are an important class of graphs for modeling many real applications in software and information visualization. In this paper, we investigate area requirements for drawing hierarchically planar graphs regarding two different drawing standards. Firstly, we show an exponential lower bound for the area needed for straight-line drawing of hierarchically planar graphs. The lower bound holds even for s- t hierarchical graphs without transitive arcs, in contrast to the results for upward planar drawing. This motivates our investigation of another drawing standard grid visibility representation, as a relaxation of straight-line drawing. An application of the existing results from upward drawing can guarantee a quadric drawing area for grid visibility representation but does not necessarily guarantee the minimum drawing area. Motivated by this, we will present a new grid visibility drawing algorithm which is efficient and guarantees the minimum drawing area with respect to a given topological embedding. This implies that the area minimization problem is polynomial time solvable restricted to the class of graphs whose planar embeddings are unique. However, we can show that the problem of area minimization of grid visibility for hierarchically planar graphs is generally NP-hard, even restricted to s- t graphs.

Complementarity and Social Networks

I investigate complementarity games played on graphs, which model negative externalities embedded in structures of interaction. On the complete graph, the traditional economic analysis applies: the number of agents playing one strategy is proportional to its payoff. I show that, in general and contrary to coordination games, the structure crucially influences the equilibria. On an important class of graphs, called bipartite graphs, the equilibria do not depend on strategies' payoffs. On certain highly asymmetric graphs, an increase in the payoff of a strategy even decreases the number of agents playing this strategy. In most cases, equilibria do not maximize welfare.

Minimal reducible bounds for planar graphs

For properties of graphs P 1 and P 2 a vertex ( P 1, P 2) -partition of a graph G is a partition (V 1,V 2) of V(G) such that each subgraph G[V i] induced by V i has property P i, i=1,2 . The class of all vertex ( P 1, P 2) -partitionable graphs is denoted by P 1∘ P 2 . An additive hereditary property R is reducible if there exist additive hereditary properties P 1 and P 2 such that R= P 1∘ P 2 , otherwise it is irreducible. For a given property P a reducible property R is called a minimal reducible bound for P if P⊆ R and there is no reducible property R′ satisfying P⊆ R′⊂ R . In this paper we give a survey of known reducible bounds and we prove some new minimal reducible bounds for important classes of planar graphs. The connection between our results and Barnette's conjecture is also presented.

A SimpleO(log N) Time Parallel Algorithm for Testing Isomorphism of Maximal Outerplanar Graphs

Maximal outerplanar graphs constitute an important class of graphs, often encountered in various applications, e.g., computational geometry, robotics, etc. In this paper, we propose a parallel algorithm for testing the isomorphism of maximal outerplanar graphs. Given the ordered adjacency lists of the two graphs, the proposed algorithm tests their isomorphism inO(logN) time usingNprocessors, for graphs withNnodes on an EREW shared memory model, as well as on a hypercube arhitecture. When the adjacency matrices of the graphs are given, this algorithm can be redesigned onN2processors to run inO(logN) time.

Multiple Graph Embeddings into a Processor Array with Spanning Buses

A processor array with spanning buses (PASB) is a well-known, versatile parallel architecture. A PASB is obtained from a two-dimensional mesh by replacing each linear connection with a bus. In this paper, we show how to optimally embed multiple copies of a graph into a PASB by a labeling strategy. Our embeddings simultaneously achieve an optimal expansion, congestion, and alignment cost. First, we propose a labeling scheme for anN-node graphG, possibly disconnected, such that this labeling makes it possible to optimally embed multiple copies ofGinto anN′×N′ PASB whereN′ is divisible byN. Second, we show that many important classes of graphs admit this labeling: for example, tree, cycle, mesh of trees, and product graphs such as mesh, torus, or hypercube. Finally, we show how to optimally embed multiple copies of a graph into a multidimensional and possibly nonsquare PASB.

How Good is Recursive Bisection?

The most commonly used p-way partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal-sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the optimal bisection problem that partitions a graph into two equal-sized subgraphs to minimize the number of edges cut is NP-complete, practical RB algorithms use more efficient heuristics in place of an optimal bisection algorithm. Most such heuristics are designed to find the best possible bisection within allowed time. We show that the RB method, even when an optimal bisection algorithm is assumed, may produce a p-way partition that is very far way from the optimal one. Our negative result is complemented by two positive ones: first we show that for some important classes of graphs that occur in practical applications, such as well-shaped finite-element and finite-difference meshes, RB is within a constant factor of the optimal one "almost always." Second, we show that if the balance condition is relaxed so that each block in the p-way partition is bounded by 2n/p, where n is the number of vertices of the graph, then a modified RB finds an approximately balanced p-way partition whose cost is within an O(log p) factor of the cost of the optimal p-way partition.

Intersections of longest cycles in grid graphs

It is well-known that the largest cycles of a graph may have empty inter- section. This is the case, for example, for any hypohamiltonian graph. In the literature, several important classes of graphs have been shown to contain examples with the above property. This paper investigates a (nontrivial) class of graphs which, on the contrary, admits no such example. c 1997 John Wiley & Sons, Inc. J Graph Theory 25: 37{52, 1997

A COHESION INDEX FOR COOPERATION GRAPHS

Abstract A cooperation graph is defined as a finite, simple, connected, noncomplete, labelled graph with at least three vertices. A non-negative normalized cohesion index is defined for such graphs. The vertices with positive index form a dominating set. Explicit expressions for the determination of the indices of cohesion for some important classes of graphs are derived.