Showing papers presented at "Workshop on Graph-Theoretic Concepts in Computer Science in 1991"

Voronoi Diagrams of Moving Points in the Plane

Abstract: Consider a set of n points in the Euclidean plane each of which is continuously moving along a given trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Delaunay diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has a nearly cubic upper bound of O(n2λs(n)), where λs,(n) is the maximum length of an (n, s)-Davenport-Schinzel sequence and s is a constant depending on the motions of the point sites. In the special case of points moving at constant speed along straight lines, we get s = 4, implying an upper bound of O(n32α(n)), where α(n) is the extremely slowly-growing inverse of Ackermann 's function. Our results are a linear-factor improvement over the naive quartic bound on the number of topological events.

On Disjoint Cycles

Abstract: It is shown, that for each constant k ≥ 1, the following problems can be solved in O(n) time: given a graph G, determine whether G has k vertex disjoint cycles, determine whether G has k edge disjoint cycles, determine whether G has a feedback vertex set of size ≤ k. Also, every class \(\mathcal{G}\), that is closed under minor taking, or that is closed under immersion taking, and that does not contain the graph formed by taking the disjoint union of k copies of K3, has an \(\mathcal{O}\)(n) membership test algorithm.

Approximating Treewidth, Pathwidth, and Minimum Elimination Tree Height

Abstract: We show how the value of various parameters of graphs connected to sparse matrix factorization and other applications can be approximated using an algorithm of Leighton et al. that finds vertex separators of graphs. The approximate values of the parameters, which include minimum front size, treewidth, pathwidth, and minimum elimination tree height, are no more than O(log n) (minimum front size and treewidth) and O(log2n) (pathwidth and minimum elimination tree height) times the optimal values. In addition we examine the existence of bounded approximation algorithms for the parameters, and show that unless P = NP, there are no absolute approximation algorithms for them.

Monadic Second-Order Evaluations on Tree-Decomposable Graphs

Abstract: Every graph generated by a hyperedge replacement graph-grammar can be represented by a tree, namely the derivation tree of the derivation sequence that produced it. Certain functions on graphs can be computed recursively on the derivation trees of these graphs. By using monadic second-order logic and semiring homomorphisms, we describe in a single formalism a large class of such functions. Polynomial and even linear algorithms can be constructed for some of these functions. We unify similar results obtained by Takamizawa et al. (1982), Bern et al. (1987), Arnborg et al. (1991) and Habel et al. (1989).

Finding Minimally Weighted Subgraphs

Abstract: Let H be a fixed graph of tree-width w. We study the problems of finding a minimally weighted weak H-subgraph, resp., a minimally weighted induced H-subgraph in an edge-weighted graph G on n nodes. We present an O(nw+1)-algorithm for the weak subgraph problem. If the maximal degree of G is bounded, the algorithm above can be modified to an O(nw+1)-algorithm for the induced subgraph problem.

Algebraic Approach to Graph Transformation Based on Single Pushout Derivations

Abstract: The Berlin approach to graph transformation, which uses double pushout derivations in the category of graphs and total graph morphisms, is modified using single pushout derivations in the category of graphs and partial graph morphisms. It is shown that the single pushout approach generalizes the classical approach in the sense that all double pushout derivations correspond to single pushout transformations but not vice versa.

The Complexity of Approximating the Class Steiner Tree Problem

Abstract: Given a connected, undirected distance graph with required classes of nodes and optional Steiner nodes, find a shortest tree containing at least one node of each required class. This problem called {\sc Class Steiner Tree} ist NP-hard and therefore we are dependent on approximation. In this paper, we investigate various restrictions of the problem comparing their complexities with respect to approximability. A main result is that for an input of trees without Steiner nodes and unit edges only, {\sc Class Steiner Tree} ist as hard to approximate as {\sc minimum set cover}, for which no constant approximation is known, too. Further we prove that if this restricted version has an approximation scheme, all members of the optimization problem class MAX SNP do.

Nondeterministic Control Structures for Graph Rewriting Systems

Abstract: The work reported here is part of the IPSEN3 project whose very goal is the development of an Integrated Project Support ENvironment. Within this project directed, attributed, node- and edge- labeled graphs (diane graphs) are used to model the internal structure of software documents and PROgrammed Graph REwriting SyStems are used to specify the operational behavior of document processing tools like syntax-directed editors, static analyzers, or incremental compilers and interpreters. Recently a very high-level language, named PROGRESS, has been developed to support these activities. This language offers its users a convenient, partly textual, partly graphical concrete syntax and a rich system of consistency checking rules (mainly type compatibility rules) for the underlying calculus of programmed diane-graph rewriting systems.

A New Upper Bound on the Complexity of the All Pairs Shortest Path Problem

Abstract: The all pairs shortest path (APSP) problem is to compute shortest paths between all pairs of vertices of a directed graph with non-negative edge costs. We present an algorithm that computes shortest distances between all pairs of vertices, since shortest paths can be computed easily as by-products as in most other algorithms. It is well-known that the time complexity of (n, n)-distance matrix multiplication (DMM) is asymptotically equal to that of the APSP problem for a graph with n vertices. See Aho, ttopcroft and Ullman [1] for example. Based on this fact, Fredma~ [5] invented an algorithm for DMM of O(n3(log log n/log n) 1/3) time, which is o(n3), meaning that the APSP problem can be solved with the same time complexity. In the average case, Moffat and Takaoka [6] solved this problem with O(n 9 log n) expected time. Our algorithm in this paper solves DMM in O(n3(loglog n/log n) ~/2) time, meaning that the APSP problem can be solved with the same time complexity. This is an asymptotic improvement of Fredman's algorithm by the factor of (logn/loglogn) 1Is. Another merit of our algorithm is that it is simple and easy to implement, whereas Fredman's algorithm is complicated and difficult to implement. Also a possible parallel implementation is mentioned. The base of logarithm is assumed to be two in this paper and fractions are rounded up if necessary.

On Complexity of Some Chain and Antichain Partition Problems

Abstract: In the paper we deal with computational complexity of a problem C k (respectively A k ) of a partition of an ordered set into minimum number of at most k-element chains (resp. antichains). We show that C k , k ≥ 3, is NP-complete even for N-free ordered sets of length at most k, C k and A k are polynomial for series-paralel orders and A k is polynomial for interval orders. We also consider related problems for graphs.

Optimal Embedding of Complete Binary Trees into Lines and Grids

Abstract: We consider several graph embedding problems which have applications in parallel and distributed computing and which have been unsolved so far. Our major result is that the complete binary tree can be embedded into the square grid of the same size with almost optimal dilation (up to a very small factor). To achieve this, we first state an embedding of the complete binary tree into the line with optimal dilation.

Induced subgraph isomorphism for cographs is NP-complete

Abstract: We prove by a reduction from 3SAT that the INDUCED SUBGRAPH ISOMORPHISM problem remains NP-complete when restricted to cographs (P4-free graphs).

The VLSI layout problem in various embedding models

Abstract: In 1984 Kramer and van Leeuwen proved a fundamental complexity result of VLSI layout theory. They showed that the so-called General Layout Problem, i.e. the problem of embedding a graph into a grid of minimum area is NP-hard, even for connected (but not necessarily planar) graphs.

On the complexity of some coloring games

Abstract: In this paper we consider the following game: players must alternately color the lowest numbered uncolored vertex of a given graph G = ({1,2, ..., n}, E) with a color, taken from a given set C, such that never two adjacent vertices are colored with the same color. In one variant, the first player which is unable to move, loses the game. In another variant, player 1 wins the game if and only if the game ends with all vertices colored. We show that for both variants, the problem to determine whether there is a winning strategy for player 1 is PSPACE-complete for any C with |C| ≥ 3, but the problems are solvable in O(n + eα(e, n)), and O(n + e) time, respectively, if |C| = 2. We also give polynomial time algorithms for the problems with certain restrictions on the graphs and orderings of the vertices. We give some partial results for the versions, where no order for coloring the vertices is specified.

A Language for Generic Graph-Transformations

Abstract: We define a class of graphs that represent object-oriented databases. Queries, updates and restructurings in such databases are performed by transformations of these graphs. We therefore first define five transformation-operations on this graph-class. Then we define generic transformations as a natural generalization of the concept of BP-completeness, originally defined for the relational database model. Our main result states that the language consisting of the five transformation-operations expresses exactly the generic transformations.

A Generalized Encryption Scheme Based on Random Graphs

Abstract: A probabilistic encryption scheme is generalized in such a way that an encryption is sometimes ambiguous (the same object might be an encryption of both 0 and 1), but a probability of this event is very low. Such a generalization is sufficient for a large number of applications.

Line Digraph Iterations and Spread Concept - with Application to Graph Theory, Fault Tolerance, and Routing

Abstract: This paper is concerned with the spread concept, line digraph iterations, and their relationship. A graph has spread (m, k, l) if for any m + 1 distinct nodes x,y1,..., y m and m positive integers r1,..., r m such that ⌆iri= k, there exist k node-disjoint paths of length at most l from x to the y i , where r i of them end at y i . This concept contains, and is related to, many important concepts used in communications and graph theory. The line digraph of a digraph G(V, E) is the digraph L(G) where nodes represent the edges of G and there is an edge (x,y) in L(G) if and only if x represents the edge (u, v) in G and y represents the edge (v,w) in G for some u, v,w e V(G). Many useful graphs, like the de Bruijn and Kautz digraphs, can be generated by line digraph iterations. We prove an optimal general theorem about the spreads of digraphs generated by line digraph iterations. Then we apply it to the de Bruijn and Kautz digraphs to derive optimal bounds on their spreads, which improve, re-prove, or resolve previous results and open questions on the connectivity, diameter, k-diameter, diameter vulnerability, and some other issues related to length-bounded disjoint paths, of these two graphs.

On the Crossing Number of the Hypercube and the Cube Connected Cycles

Abstract: We prove tight bounds for the crossing number of the n-dimensional hypercube and cube connected cycles (CCC) graphs.

Dynamic Algorithms for Shortest Paths in Planar Graphs

Abstract: We propose data structures for maintaining shortest path in planar graphs in which the weight of an edge is modified. Our data structures allow us to compute after an update the shortest-path tree rooted at an arbitrary query node in time O(n√log log n) and to perform an update in O((log n)3). Our data structure can be also applied to the problem of maintaining the maximum flow problem in an s − t planar network.

Attributed Elementarz Programmed Graph Grammars

Abstract: A new mechanism for generating graph languages is introduced which is based on the controlled rewriting of graphs using only six elementary types of graph productions, namely the addition, the deletion and the renaming of a node or an edge Although these elementary graph productions are acting strictly locally and no embedding transformations are needed, in the unrestricted and monotone case, elementary programmed graph grammars have the same generative power as expression graph grammars As a graph with attributes assigned to its nodes and edges is an even more useful tool for the description of certain data structures than a directed graph, attributed elementary programmed graph grammars turn out to be adequate graph rewriting systems applicable in many areas of computer science

Edge Separators for Graphs of Bounded Genus with Applications

Abstract: We prove that every n-vertex graph of genus g and maximal degree k has an edge separator of size O(√gkn). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separator to the isoperimetric number problem, graph embeddings and lower bounds for crossing numbers.

Deciding 1-solvability of distributed task is NP-hard (extended abstract)

Abstract: We present a use of graph-theoretic concepts in fault-tolerant distributed systems. The question of which distributed tasks can be solved by asynchronous protocols in the presence of crash failures has been investigated in recent years. Some of these studies investigate the solvability of specific tasks, providing both positive and negative results. Given a distributed task, we associate with it certain graphs, that enable us to determine whether this task can be solved in the presence of on faulty processor ([BMZ]), and in this paper we further use this approach to study the following problem: Given a task T for N processors, how difficult is it to determine whether T can be solved in the presence of one faulty processor? We show that this problem is NP-hard for any fixed N > 3. 1 This research was supported in part by Technion V.P.R. Funds Wellner Research Fund and Loewengart Research Fund, by the I. Goldberg Fund for Electronics Research, and by the Foundation for Research in Electronics, Computers and Communications, administrated by the Israel Academy of Sciences and Humanities.

Graph Rewriting Systems and their Application to Network Reliability Analysis

Abstract: We propose a new kind of Graph Rewriting Systems (GRS) that provide a theoretical foundation for using the reduction methods to analyze network reliability, and give the critical pair lemma in this paper.

Logic Arrays for Interval Indicator Functions

Abstract: Programmable logic arrays (PLA) are highly regular circuits which compute suitable Boolean functions. PLAs are a popular circuit design especially in microcontrollers. PLA folding is a technique for minimizing the area of such PLA. Folding can be formulated in purely graph-theoretic terms. In this paper, we study PLA for a very special class of Boolean functions, namely indicator functions of intervals of binary numbers. As a possible application, we can imagine PLA which shall supervise that an input value lies in a prescribed tolerance interval. From a mathematical point of view, the connection of these interval indicator functions with threshold graphs and cographs is interesting. Our study naturally leads to the folding problem in cographs which is shown to be solvable in linear time. In sections 1-2 we introduce the basic notions. In section 3 we prove a theorem on minimal normal forms of interval indicator functions, thereby motivating the folding problem on cographs. A linear time algorithm solving this problem is derived in section 4. Finally, in section 5, we mention some related results and possible directions of further research.

Tight bounds for the rectangular art gallery problem

Abstract: Consider a rectangular art gallery, subdivided into n rectangular rooms; any two adjacent rooms have a door connecting them. We show that ⌈n/2⌋ guards are always sufficient to protect all rooms in a rectangular art gallery; furthermore, their positioning can be determined in O(n) time. We show that the optimal positioning of the guards can be determined in linear time. We extend the result by proving that in an arbitrary orthogonal art gallery (not necessarily convex, possibly having holes) with n rectangular rooms and k walls, ⌈(n + k)/2⌋ guards are always sufficient and occasionally necessary to guard all the rooms in our gallery. A linear time algorithm to find the positioning of the guards is obtained.

Copmlete Problems for Logspace Involving Lexicographic First Paths in Graphs

Abstract: It is shown that the problem of deciding whether a given vertex is on the lexicographic first path of some digraph, starting at some other specified vertex, is complete for deterministic logspace via projection translations: such translations are extremely weak forms of reductions. Other related problems involving constrained versions of the lexicographically first path problem in both digraphs and graphs are also shown to be similarly complete. The methods used to prove completeness involve the consideration of decision problems as sets of finite structures satisfying certain logical formulae.

Optimal Vertex Ordering of a Graph and its Application to Symmetry Detection

Abstract: We consider in this paper the problem of optimal vertex ordering of a graph. The vertex ordering and an optimality measure are defined. It is proved that the optimal ordering problem can be transformed into the well-known minimum-weight spanning tree problem. Some properties of optimal vertex orderings are investigated. And finally, the application of the optimal vertex ordering technique to an algorithm for detecting the symmetry of polyhedra is discussed in some detail.

Two polynomial problems in PLA folding

Abstract: Block-folding and variable-folding are widely used techniques for reducing the physical area of Programmed Logic Arrays (PLA). Both block- and variable-folding problems are known to be NP-hard. We define the compatibility graph of a PLA as the complement of its column-disjoint graph, and prove that both block-folding and variable-folding can be solved in polynomial time on PLA whose compatibility graph does not contain a claw or a (K5 − e) as induced subgraph.