Showing papers in "SIAM Journal on Discrete Mathematics in 1993"
TL;DR: This paper presents an algorithm to solve the discrete logarithm problem forGF ( p) with heuristic expected running time L_p [ 1/3; 3^{2/3}] and for umbers of a special form, there is an asymptotically slower but more practical version of the algorithm.
Abstract: Recently, several algorithms using number field sieves have been given to factor a number n in heuristic expected time $L_n [1/3; c]$, where \[ L_n [ v ;c ] = \exp \left\{ ( c + o ( 1 ) ) ( \log n )^v ( \log \log n )^{1 - v } \right\} \] for $n \to \infty $.This paper presents an algorithm to solve the discrete logarithm problem for $GF ( p )$ with heuristic expected running time $L_p [ 1/3; 3^{2/3}]$. For umbers of a special form, there is an asymptotically slower but more practical version of the algorithm.
351 citations
TL;DR: Using the Minkowski addition of Newton polytopes, the authors show that the following problem can be solved in polynomial time for any finite set of polynomials $\mathcal{T} \subset K [ x_1, \ldots,x_d ]$, where d is fixed.
Abstract: This paper deals with a problem from computational convexity and its application to computer algebra. This paper determines the complexity of computing the Minkowski sum of k convex polytopes in $\mathbb{R}^d $, which are presented either in terms of vertices or in terms of facets. In particular, if the dimension d is fixed, the authors obtain a polynomial time algorithm for adding k polytopes with up to n vertices. The second part of this paper introduces dynamic versions of Buchberger’s Grobner bases algorithm for polynomial ideals. Using the Minkowski addition of Newton polytopes, the authors show that the following problem can be solved in polynomial time for any finite set of polynomials $\mathcal{T} \subset K [ x_1 , \ldots ,x_d ]$, where d is fixed: Does there exist a term order $\tau $ such that $\mathcal{T}$ is a Grobner basis for its ideal with respect to $\tau $? If not, find an optimal term order for $\mathcal{T}$ with respect to a natural Hilbert function criterion.
222 citations
TL;DR: It is shown that the pathwidth of a cograph equals its treewidth, and a linear time algorithm to determine the path Width and build a corresponding path-decomposition is given.
Abstract: It is shown that the pathwidth of a cograph equals its treewidth, and a linear time algorithm to determine the pathwidth of a cograph and build a corresponding path-decomposition is given.
198 citations
TL;DR: The authors use a novel potential function argument to show that in the worst case M = ( \frac{4}{27} + o ( 1 ) )n^3 $, the issue was to obtain a polynomial bound.
Abstract: A token located at some vertex $v $ of a connected, undirected graph G on n vertices is said to be taking a “random walk” on G if, whenever it is instructed to move, it moves with equal probability to any of the neighbors of $v $. The authors consider the following problem: Suppose that two tokens are placed on G, and at each tick of the clock a certain demon decides which of them is to make the next move. The demon is trying to keep the tokens apart as long as possible. What is the expected time M before they meet?The problem arises in the study of self-stabilizing systems, a topic of recent interest in distributed computing. Since previous upper bounds for M were exponential in n, the issue was to obtain a polynomial bound. The authors use a novel potential function argument to show that in the worst case $M = ( \frac{4}{27} + o ( 1 ) )n^3 $.
169 citations
TL;DR: The authors determine the algorithmic complexity of domination and variants on cocomparability graphs, a class of perfect graphs containing both the interval and the permutation graphs.
Abstract: The authors determine the algorithmic complexity of domination and variants on cocomparability graphs, a class of perfect graphs containing both the interval and the permutation graphs. Minimum dominating, total dominating, connected dominating, and independent dominating sets can be constructed in polynomial time. On the other hand, DOMINATING CLIQUE and MINIMUM DOMINATING CLIQUE remain NP-complete on cocomparability graphs.
157 citations
TL;DR: It is shown that every 3-connected planar graph G can be represented as a collection of circles, one circle representing each vertex and each face, so that, for each edge of G, the four circles representing the two endpoints and the two neighboring faces meet at a point, and furthermore the vertex-circles cross the face- circles at right angles.
Abstract: This paper shows that every 3-connected planar graph G can be represented as a collection of circles, one circle representing each vertex and each face, so that, for each edge of G, the four circles representing the two endpoints and the two neighboring faces meet at a point, and furthermore the vertex-circles cross the face-circles at right angles. This extends a result of W. Thurston [The Geometry and Topology of Three Manifolds, unpublished] and, independently, Andreev. From this we deduce two corollaries: (1) The partial order formed by taking the vertices, edges, and bounded faces of G, ordered by inclusion, is a circle order; (2) One can represent G and its dual simultaneously in the plane with straight-line edges so that the edges of G cross the dual edges at right angles. This answers a question first asked by W. Tutte [Proc. LMS, 13 (3) (1963), pp. 743–768].
138 citations
TL;DR: The edge domination problem is NP-complete for planar bipartite graphs, their subdivision, line, and total graphs, perfect claw-free graphs, and planar cub...
Abstract: Let $G = ( V,E )$ be a finite undirected graph with n vertices and m edges. A minimum edge dominating set of G is a set of edges D, of smallest cardinality $\gamma ' ( G )$, such that each edge of $E - D$ is adjacent to some edge of D. Let $S( G )$ be the subdivision graph of G and let $T( G )$ be the total graph of G. Let $\alpha ( G )$ be the stability number of G (cardinality of a largest stable set) and let $\alpha _2 ( G )$ be the 2-stability number of G (cardinality of a largest set of vertices in G, no two of which are joined by a path of length 2 or less). The following results are obtained. For any $G,\gamma' ( S ( G ) ) + \alpha _2 ( G ) = n$ and $2\gamma ' ( T ( G ) ) + \alpha ( T ( G ) ) = n + m$ or $n + m + 1$. Also, for any depth-first search tree S of $G,\gamma ' ( S )/2\leqq \gamma ' ( G )\leqq 2\gamma ' (S)$, and these bounds are tight.The edge domination problem is NP-complete for planar bipartite graphs, their subdivision, line, and total graphs, perfect claw-free graphs, and planar cub...
133 citations
TL;DR: It is proved that every triangle-free graph on n \geq 4 vertices has at most $2 n /2 $ or $5 \cdot 2^{( n - 5 )/2} $ independent sets maximal under inclusion, whether n is even or odd.
Abstract: In this paper, it is proved that every triangle-free graph on $n \geq 4$ vertices has at most $2^{n /2} $ or $5 \cdot 2^{( n - 5 )/2} $ independent sets maximal under inclusion, whether n is even or odd In each case, the extremal graph is unique If the graph is a forest of odd order, then the upper bound can be improved to $2^{( n - 1 )/2} $
105 citations
TL;DR: It is proved that the multiparty communication complexity problems are equivalent to certain hypergraph properties and thereby establish the connections among a large number of combinatorial and computational aspects of hypergraphs or Boolean functions.
Abstract: The multiparty communication complexity concerns the least number of bits that must be exchanged among a number of players to collaboratively compute a Boolean function $f ( x_1 , \ldots ,x_k )$, while each player knows at most t inputs for some fixed $t < k$. The relation of the multiparty communication complexity to various hypergraph properties is investigated. Many of these properties are satisfied by random hypergraphs and can be classified by the framework of quasi randomness. Namely, many disparate properties of hypergraphs are shown to be mutually equivalent, and, furthermore, various equivalence classes form a natural hierarchy. In this paper, it is proved that the multiparty communication complexity problems are equivalent to certain hypergraph properties and thereby establish the connections among a large number of combinatorial and computational aspects of hypergraphs or Boolean functions.
87 citations
TL;DR: A natural class of random pairs is considered and the random pair $( X,Y )$ is balanced if $\hat \mu = \hat \eta $, which holds for all balanced random pairs.
Abstract: $( X,Y )$ is a pair of random variables distributed over a support set S. Person $P_X $ knows X, person $P_Y $ knows Y, and both know S. Using a predetermined protocol, they exchange binary messages for $P_Y $ to learn X. $P_X$ may or may not learn Y. The m-message complexity $\hat C_m $ is the number of information bits that must be transmitted (by both persons) in, the worst case if only m messages are allowed. $\hat C_\infty $ is the number of bits required when there is no restriction on the number of messages exchanged.A natural class of random pairs is considered. $\hat \mu $ is the maximum number of X values possible with a given Y value. $\hat \eta $ is the maximum number of Y values possible with a given X value. The random pair $( X,Y )$ is balanced if $\hat \mu = \hat \eta $. The following hold for all balanced random pairs. One-way communication requires at most twice the minimum number of bits: $\hat C_1 \leqq 2\hat C_\infty + 1$. This bound is almost tight: For every $\alpha $, there is a ba...
75 citations
TL;DR: The tessellation of the plane given by square cells of equal size can be considered the Cayley graph of the free abelian group of rank 2, which has polynomial growth.
Abstract: The tessellation of the plane given by square cells of equal size can be considered the Cayley graph of the free abelian group of rank 2. This group has polynomial growth. The theorems of Moore [Symposium on Applied Mathematics, Vol. XIV, American Mathematical Society, Providence, Rhode Island, 1962, pp. 17–33] and Myhill [Proceedings of the American Mathematical Society, 14 (1963), pp. 685–686] on the existence of Garden of Eden configurations for an automaton defined on such a graph are extended to Cayley graphs of groups whose growth function is not exponential. Examples are given of Cayley graphs of groups of exponential growth for which these theorems do not hold.
TL;DR: It is shown how to compactly encode the entire lattice and it is shown that the set of solutions to the min-cost flow problem forms a sublattice in the presented lattice.
Abstract: Flow in planar graphs has been extensively studied, and very efficient algorithms have been developed to compute max-flows, min-cuts, and circulations. Intimate connections between solutions to the planar circulation problem and with “consistent” potential functions in the dual graph are shown. It is also shown that the set of integral circulations in a planar graph very naturally forms a distributive lattice whose maximum corresponds to the shortest path tree in the dual graph. Further characterized is the lattice in terms of unidirectional cycles with respect to a particular face called the root face. It is shown how to compactly encode the entire lattice and it is also shown that the set of solutions to the min-cost flow problem forms a sublattice in the presented lattice.
TL;DR: It is shown that, for any basis B of transpositions for S_n, there is a Hamilton cycle in Cay( B:S_n )$ that includes every edge of $M_b $ in Cay, and that it is possible to generate all permutations of $1,2, \ldots ,n$ byTranspositions in B so that every other transposition is b.
Abstract: Let B be a basis of transpositions for $S_n $ and let Cay$( B:S_n )$ be the Cayley graph of $S_n $ with respect to B. It was shown by Kompel’makher and Liskovets [Kibemetica, 3 (1975), pp. 17–21] that Cay$( B:S_n )$ is Hamiltonian. This result is extended as follows. Note that every transposition b in B induces a perfect matching $M_b $ in Cay$( B:S_n )$. It is shown here when $n > 4$ that, for any $b \in B$, there is a Hamilton cycle in Cay$( B:S_n )$ that includes every edge of $M_b $. That is, for $n > 4$, for any basis B of transpositions of $S_n $, and, for any $b \in B$, it is possible to generate all permutations of $1,2, \ldots ,n$ by transpositions in B so that every other transposition is b.
TL;DR: This paper shows that the depth in the asymmetric case (i.e., symbols from the alphabet do not occur with the same probability) is asymptotically normally distributed.
Abstract: Digital tries occur in a variety of computer and communication algorithms, including symbolic manipulations, compiling, comparison-based searching and sorting, digital retrieval techniques, algorithms on strings, file systems, codes, and communication protocols. The depth of the PATRICIA trie in a probabilistic framework is studied. The PATRICIA trie is a digital tree in which nodes that would otherwise have only one branch have been collapsed into nodes having more than one branch. Because of this characteristic, the depth of the PATRICIA trie provides a measure on the compression of the keys stored in the trie. Here, n independent keys that are random strings of symbols from a V-ary alphabet are considered. This model is known as the Bernoulli model. This paper shows that the depth in the asymmetric case (i.e., symbols from the alphabet do not occur with the same probability) is asymptotically normally distributed. In the symmetric case, which surprisingly proved to be more difficult, the limiting gener...
TL;DR: It is shown that these four problems are NP-complete in split graphs with degree constraints and linear time algorithms for them are given in a strongly chordal graph when a strong elimination order is given.
Abstract: In a graph $G = ( V,E ),E [ v ]$ denotes the set of edges in the subgraph induced by $N [ v ] \equiv \{ v \} \cup \{ u \in V:uv \in E \}$. The neighborhood-covering problem is to find the minimum cardinality of a set C of vertices such that $E = \cup \{ E [ v ]:v \in C \}$. The neighborhood-independence problem is to find the maximum cardinality of a set of edges in which there are no two distinct edges belonging to the same $E [ v ]$ for any $v \in V$. Two other related problems are the clique-transversal problem and the clique-independence problem. It is shown that these four problems are NP-complete in split graphs with degree constraints and linear time algorithms for them are given in a strongly chordal graph when a strong elimination order is given.
TL;DR: An algorithm for finding a longest path in a complete m-partite $( m \geq 2 )$ digraph with n vertices with time $O ( n^3 )$ in case of testing only the existence of a Hamiltonian path and finding it if one exists.
Abstract: A digraph obtained by replacing each edge of a complete m-partite graph with an arc or a pair of mutually opposite arcs with the same end vertices is called a complete m-partite digraph. An $O ( n^3 )$ algorithm for finding a longest path in a complete m-partite $( m \geq 2 )$ digraph with n vertices is described in this paper. The algorithm requires time $O( n^{2.5} )$ in case of testing only the existence of a Hamiltonian path and finding it if one exists. It is simpler than the algorithm of Manoussakis and Tuza [SIAM J. Discrete Math., 3 (1990), pp. 537–543], which works only for $m = 2$. The algorithm implies a simple characterization of complete m-partite digraphs having Hamiltonian paths that was obtained for the first time in Gutin [Kibernetica (Kiev), 4 (1985), pp. 124–125] for $m = 2$ and in Gutin [Kibernetica (Kiev), 1(1988), pp. 107–108] for $ m \geq 2 $.
TL;DR: The order dimension of the partially ordered set P is exactly 4 for every convex polytope M and the subposet of P determined by the vertices and faces is critical in the sense that deleting any element leaves a poset of dimension 3.
Abstract: With a convex polytope ${\text{M}}$ in $\mathbb{R}^3$, a partially ordered set ${\text{P}}_{\text{M}} $ is associated whose elements are the vertices, edges, and faces of ${\text{M}}$ ordered by inclusion. This paper shows that the order dimension of ${\text{P}}_{\text{M}} $ is exactly 4 for every convex polytope ${\text{M}}$. In fact, the subposet of ${\text{P}}_{\text{M}} $ determined by the vertices and faces is critical in the sense that deleting any element leaves a poset of dimension 3.
TL;DR: In this paper, it was shown that the greedy algorithm optimizes all linear objective functions if and only if the problem structure (phrased in terms of either accessible set systems or hereditary languages) is a matroid embedding.
Abstract: The authors present exact characterizations of structures on which the greedy algorithm produces optimal solutions. Our characterization, which are called matroid embeddings, complete the partial characterizations of Rado [A note on independent functions, Proc. London Math. Soc., 7 (1957), pp. 300–320], Gale [Optimal assignments in an ordered set, J. Combin. Theory, 4 (1968), pp. 176–180], and Edmonds [Matroids and the greedy algorithm, Math. Programming, 1 (1971), pp. 127–136], (matroids), and of Korte and Lovasz [Greedoids and linear object functions, SIAM J. Alg. Discrete Meth., 5 (1984), pp. 239–248] and [Mathematical structures underlying greedy algorithms, in Fundamentals of Computational Theory, LNCS 177, Springer-Verlag, 1981, pp. 205–209] (greedoids). It is shown that the greedy algorithm optimizes all linear objective functions if and only if the problem structure (phrased in terms of either accessible set systems or hereditary languages) is a matroid embedding. An exact characterization of the ...
TL;DR: The author proves that the forwarding index problem is NP-complete even if the diameter of the graph is 2, thereby answering a question of F. Chung et al. concerning the complexity of the problem.
Abstract: The forwarding index problem is, given a connected graph G and an integer k, finding a way of connecting each ordered pair of vertices by a path so that every vertex is an internal point of at most k such paths. Such a problem arises in the design of communication networks and parallel architectures, a model of parallel computation being represented by a network of processors or machines processing and forwarding(synchronous) messages to each other and a physical constraint on the number of messages that can be processed by a single machine. In this paper, the author proves that the forwarding index problem is NP-complete even if the diameter of the graph is 2, thereby answering a question of F. Chung et al. [IEEE Trans. Inform. Theory, 33 (1987), pp. 224–232] concerning the complexity of the problem.
TL;DR: The author presents a sequence of linear-time, bounded-space, on-line, bin-packing algorithms that are based on the “HARMONIC” algorithms introduced by Lee and Lee, which guarantee the worst case performance of H_k, whereas they only use O ( \log k ) instead of k active bins.
Abstract: The author presents a sequence of linear-time, bounded-space, on-line, bin-packing algorithms that are based on the “HARMONIC” algorithms ${\text{H}}_k $ introduced by Lee and Lee [J. Assoc. Comput. Mach., 32 (1985), pp. 562–572]. The algorithms in this paper guarantee the worst case performance of ${\text{H}}_k $, whereas they only use $O ( \log \log k )$ instead of k active bins. For $k\geqq 6$, the algorithms in this paper outperform all known heuristics using k active bins. For example, the author gives an algorithm that has worst case ratio less than 17/10 and uses only six active bins.
TL;DR: The technique proposed here is to determine the induced cycle structure of a signed permutation by the number of fixed vertices or fixed edges of asigned permutation in the cyclic group generated by a signedpermutation of given type.
Abstract: The hyperoctahedral group B{sub n} is treated as the automorphism group of the n-dimensional hypercube, denoted Q{sub n}, which is nowadays understood to be a graph on 2{sup n} vertices. It is well-known that B{sub n} can be represented by the group of signed permutations. In other words, any signed permutation induces a permutation on the vertices of Q{sub n} which preserves adjacencies. Moreover, signed permutations also a permutation group on the edge of Q{sub n}, denoted H{sub n}. We study the cycle structures of both B{sub n} and H{sub n}. The technique proposed here is to determine the induced cycle structure of a signed permutation by the number of fixed vertices or fixed edges of a signed permutation in the cyclic group generated by a signed permutation of given type. Here we directly define the type of a signed permutation by a double partition based on its signed cycle decomposition. In this way, we obtain explicit formulas for the number of induced cycles on vertices as well as edges of Q{sub n} of a signed permutation in terms of its type. By further exploring the connection between cycle indices and the structure of fixed points, we obtain the cyclemore » indices of both B{sub n} and H{sub n}. Our formula for the cycle index of B{sub n}is much more natural and considerably simpler than that of Harrison and High. Meanwhile, the cycle structure of H{sub n} seems to have been untouched before, although it is well motivated by nonisomorphic edge colorings of Q{sub n} as well as by the recent interest in symmetries of computer networks.« less
TL;DR: This paper introduces a new class of valid inequalities for the polytope associated with the minimum cost 2-edge connected subgraph problem, and necessary and sufficient conditions for these inequalities to be facet-inducing for thispolytope are given.
Abstract: The problem of designing communication networks that can survive the loss of any single link is studied. Such problems can be formulated as minimum cost 2-edge connected subgraph problems in a complete graph. The linear programming cutting plane approach has been used effectively for related problems in [Schwerpunktprogramm der Deutschen Forschungsgemeinschaft, Anwendungsbezogene Optimierung and Steuerung, Report No. 188, 1989], where problem-specific cutting planes that define facets of the underlying integer polyhedra are used. This paper introduces a new class of valid inequalities for the polytope associated with the minimum cost 2-edge connected subgraph problem, and necessary and sufficient conditions for these inequalities to be facet-inducing for this polytope are given.
TL;DR: It is noted that three-colored triangulatable graphs are always planar, and this fact is modified to modify Kannan and Warnow’s algorithm to obtain an algorithm that uses both linear time and linear space.
Abstract: Kannan and Warnow [Triangulating Three-Colored Graphs, Proc. 2nd SODA, 1991, pp. 337–343 and SIAM J. Discrete Math., 5 (1992), pp. 249–258] describe an algorithm to decide whether a three-colored graph can be triangulated so that all the edges connect vertices of different colors. This problem is motivated by a problem in evolutionary biology. Kannan and Warnow have two implementation strategies for their algorithm: one uses slightly superlinear time, while the other uses linear time but quadratic space. We note that three-colored triangulatable graphs are always planar, and we use this fact to modify Kannan and Warnow’s algorithm to obtain an algorithm that uses both linear time and linear space.
TL;DR: Several constructions are presented for extending a bounded-to-one sliding-block code to a bounded to-one surjection onto its range, while preserving nice properties of the original code.
Abstract: Several constructions are presented for extending a bounded-to-one sliding-block code to a bounded-to-one surjection onto its range, while preserving nice properties of the original code.
TL;DR: The authors use the recently developed theory of majorization and Schur convexity with respect to partially ordered sets to study optimal multipartitions for the above problem and apply the results to construct a class of counterexamples to a recent conjecture.
Abstract: In a $( t,n,m )$-multipartitioning problem, t lists of $nm$ ordered numbers are partitioned into n sets, where each set contains m numbers from each list. The goal is to maximize some objective function that depends on the sum of the elements in each set and is called the partition function. The authors use the recently developed theory of majorization and Schur convexity with respect to partially ordered sets to study optimal multipartitions for the above problem. In particular, they apply the results to construct a class of counterexamples to a recent conjecture of Du and Hwang, which asserts that (classic) Schur convex functions can be characterized as the partition functions for $( 1,n,m )$-multipartitioning problems having monotone optimal solutions.
TL;DR: A surprisingly efficient algorithm for testing whether a linear combination of suppressed cells is an invariant, which exploits a linear algebraic structure of directed and undirected cycles in a mixed graph induced by a given table.
Abstract: To protect sensitive information in a cross tabulated table, it is a common practice to suppress some of the cells in the table. A linear combination of suppressed cells is called a linear invariant if the combination has a unique feasible value. Intuitively, the information contained in an invariant is not protected even though the values of the suppressed cells are not disclosed. This paper gives a surprisingly efficient algorithm for testing whether a linear combination of suppressed cells is an invariant. In sequential computation, the algorithm runs in optimal linear time. In parallel computation, the algorithm runs in polylogarithmic time using a polynomial number of processors on a parallel random access machine. The algorithm exploits a linear algebraic structure of directed and undirected cycles in a mixed graph induced by a given table. This new structure also plays a crucial role in subsequent papers on other aspects of detecting and protecting sensitive information in a cross tabulated table.
TL;DR: It is shown that all degree-4 Borel Cayley graphs can also be represented by the more restrictive chordal rings (CR) through a constructive proof.
Abstract: There is a continuing search for dense $( \delta ,D )$ interconnection graphs, that is, regular, undirected, degree $\delta $ graphs with diameter D and having a large number of nodes. Cayley graphs formed by Borel subgroups currently contribute to some of the densest known $( \delta = 4,D )$ graphs for a range of D [1]. However, the group theoretic representation of these graphs makes the development of efficient routing algorithms difficult. In an earlier report, it was shown that all Cayley graphs have generalized chordal ring (GCR) representations [2]. In this paper, it is shown that all degree-4 Borel Cayley graphs can also be represented by the more restrictive chordal rings (CR) through a constructive proof. A step-by-step algorithm to transform any degree-4 Borel Cayley graph into a CR graph is provided. Examples are used to illustrate this concept.
TL;DR: In this article, a parallel algorithm for finding an independent set of size at least $n 2 / (2m + n) is presented, where n is the number of vertices in the graph.
Abstract: Every graph with n vertices and m edges has an independent set containing at least $n^2 / (2m + n)$ vertices. This paper presents a parallel algorithm that finds an independent set of this size and...
TL;DR: A fast parallel algorithm for the recognition of ultrametrics is presented and its time-processor product is of the same order as the time bound of the known sequential algorithm of Culberson and Rudnicki.
Abstract: A fast parallel algorithm for the recognition of ultrametrics is presented. Its time-processor product is of the same order as the time bound of the known sequential algorithm of Culberson and Rudnicki [Inform. Process. Lett., 30 (1990), pp. 215–220] (compare also [SIAM J. Disc. Math., 3 (1990), pp. 1–6] and [Quart. Appl. Math., 26 (1968), pp. 607–609]. By the same way, tree metrics also can be recognized.
TL;DR: The authors estimate $\Delta ( t,H )$ and $\Delta ' (T,H)$ for the cycle, path, complete binary tree, grid, torus, and hypercube on n vertices.
Abstract: This paper considers the following problem: Given a positive integer t and graph H, construct a graph G from H by adding a minimum number $\Delta ( t,H )$ (respectively, $\Delta ' ( t,H )$) of edges and an appropriate number of vertices, such that after removing any t vertices (respectively, t edges) from G the remaining graph contains H as a subgraph. This problem was motivated by the design of fault-tolerant interconnection networks for multiprocessor systems. The authors estimate $\Delta ( t,H )$ and $\Delta ' ( t,H )$ for the cycle, path, complete binary tree, grid, torus, and hypercube on n vertices.