Showing papers on "Split graph published in 1998"
TL;DR: A new algorithm for error-correcting subgraph isomorphism detection from a set of model graphs to an unknown input graph is proposed based on a compact representation of the model graphs that can be combined with a future cost estimation method that greatly improves its run-time performance.
Abstract: We propose a new algorithm for error-correcting subgraph isomorphism detection from a set of model graphs to an unknown input graph. The algorithm is based on a compact representation of the model graphs. This representation is derived from the set of model graphs in an off-line preprocessing step. The main advantage of the proposed representation is that common subgraphs of different model graphs are represented only once. Therefore, at run time, given an unknown input graph, the computational effort of matching the common subgraphs for each model graph onto the input graph is done only once. Consequently, the new algorithm is only sublinearly dependent on the number of model graphs. Furthermore, the new algorithm can be combined with a future cost estimation method that greatly improves its run-time performance.
378 citations
01 Jan 1998
TL;DR: This paper presents an efficient algorithm for finding a hidden clique of vertices of size k that is based on the spectral properties of the graph and improves the trivial case k ) cn log n .
Abstract: We consider the following probabilistic model of a graph on n labeled vertices First choose a random graph Gn ,1 r 2 ,and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k This question was posed independently, in various variants, by Jerrum and by Kucera In this paper we present an efficient algorithm for all k ) cn 05 , for ˇ 05 05 any fixed c ) 0, thus improving the trivial case k ) cn log n The algorithm is based on the spectral properties of the graph Q 1998 John Wiley & Sons, Inc Random Struct Alg, 13, 457)466, 1998
301 citations
TL;DR: This paper presents an efficient algorithm for all k>cn0.5(log n)0, for any fixed c>0, thus improving the trivial case k>, and based on the spectral properties of the graph.
Abstract: We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph G(n, 1/2), and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kucera. In this paper we present an efficient algorithm for all k>cn0.5, for any fixed c>0, thus improving the trivial case k>cn0.5(log n)0.5. The algorithm is based on the spectral properties of the graph. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 457–466, 1998
285 citations
TL;DR: A polynomial time algorithm is given to solve the problem when the input graph is an interval graph and it is argued that the complexity of certain other modifications of the problem are likely to be difficult to classify.
173 citations
TL;DR: A unified framework for characterizations of graphs with maximum neighborhood orderings in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal is given.
Abstract: Recently in several papers, graphs with maximum neighborhood orderings were characterized and turned out to be algorithmically useful. This paper gives a unified framework for characterizations of those graphs in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal. These graphs are dual (in the sense of hypergraphs) to chordal graphs. By using the hypergraph approach in a systematical way new results are obtained, some of the old results are generalized, and some of the proofs are simplified.
141 citations
TL;DR: Using voltage graphs, a family of vertex-transitive non-Cayley graphs is constructed which shows thatvt(d,2)?(8/9)(d+12)2 for alld of the formd=(3q?1)/2, whereqis a prime power congruent with 1 (mod 4).
113 citations
TL;DR: New classes of graphs for which the isomorphism problem can be solved in polynomial time are presented, and every such graph has a unique tree representation: the internal nodes correspond to three types of graph operations, while the leaves are basic graphs with a simple structure.
65 citations
TL;DR: This work defines a moplex as a maximal clique module the neighborhood of which is a minimal separator of the graph, and generalizes Dirac's theorem to any undirected graph: “Every non-clique graph has at least two non-adjacent moplexes”.
62 citations
Book•
13 Aug 1998
TL;DR: 1. Squaring the square 2. Knights errant 3. graphs within graphs 4. Unsymmetrical electricity 5. Algebra in graph theory 6. Symmetry in graphs 7. Graphs on spheres 8. Reconstruction 9. Planar enumeration 12. chromatic eigenvalues.
Abstract: 1. Squaring the square 2. Knights errant 3. Graphs within graphs 4. Unsymmetrical electricity 5. Algebra in graph theory 6. Symmetry in graphs 7. Graphs on spheres 8. The Cats of Cheshire 9. Reconstruction 10. Planar enumeration 11. The chromatic eigenvalues 12. In conclusion Bibliography Index
61 citations
TL;DR: It is shown that, under additional constraints, e.g., prime order and/or sparseness, graph isomorphism and minimum graph coloring become easier in the circulant case, and the techniques used take advantage of spectral techniques for their efficient computation.
55 citations
TL;DR: This paper proves a characterization of these separators in terms of the monotone adjacency sets of the vertices of the graph, numbered by the maximum cardinality search (MCS) scheme, and introduces the notion of multiplicity of a minimal vertex separator which indicates the number of different pairs of vertices separated by it.
TL;DR: A convexity lemma is proved and used to derive a simple algorithm of complexityO(mn) for recognizing median graphs.
Abstract: A hierarchy of classes of graphs is proposed which includes hypercubes, acyclic cubical complexes, median graphs, almost-median graphs, semi-median graphs and partial cubes. Structural properties of these classes are derived and used for the characterization of these classes by expansion procedures, for a characterization of semi-median graphs by metrically defined relations on the edge set of a graph and for a characterization of median graphs by forbidden subgraphs. Moreover, a convexity lemma is proved and used to derive a simple algorithm of complexityO(mn) for recognizing median graphs.
TL;DR: A simple charactrization of strongly chordal graphs is presented and it is shown that every cycle on six or more vertices has an induced triangle with exactly two edges of the triangle as the chords of the cycle.
TL;DR: This paper characterize hereditary t-perfect graphs by showing that any non-t-perfect graph contains a non--t- Perfect subdivision of K4, called a bad-K4; to prove the result, it is shown which "weakly 3-connected" graphs contain no bad- K4; as a side-product of this the authors get a polynomial time recognition algorithm.
Abstract: The richest class of t-perfect graphs known so far consists of the graphs with no so-called odd-K. Clearly, these graphs have the special property that they are hereditary t-perfect in the sense that every subgraph is also t-perfect, but they are not the only ones. In this paper we characterize hereditary t-perfect graphs by showing that any non--t-perfect graph contains a non--t-perfect subdivision of K4, called a bad-K4. To prove the result we show which "weakly 3-connected" graphs contain no bad-K4; as a side-product of this we get a polynomial time recognition algorithm.
It should be noted that our result does not characterize t-perfection, as that is not maintained when taking subgraphs but only when taking induced subgraphs.
11 Mar 1998
TL;DR: For appropriately partitioned bounded degree graphs, it is shown that the running time of the algorithm under the P-RAM computational model is of $O(1)$, which is an improvement over the previous best P- RAM complexity for this class of graphs.
Abstract: The parallel construction of maximal independent sets is a useful building block for many algorithms in the computational sciences, including graph coloring and multigrid coarse grid creation on unstructured meshes. We present an efficient asynchronous maximal independent set algorithm for use on parallel computers, for ``well partitioned'''' graphs, that arise from finite element (FE) models. For appropriately partitioned bounded degree graphs, it is shown that the running time of our algorithm under the P-RAM computational model is of $O(1)$, which is an improvement over the previous best P-RAM complexity for this class of graphs. We present numerical experiments on an IBM SP, that confirm our P-RAM complexity model is indicative of the performance one can expect with practical partitions on graphs from FE problems.
08 Nov 1998
TL;DR: It is shown that when p<(1-/spl epsiv/) In n/|S|, an independent set of size |S| cannot be recovered, unless NP/spl sube/BPP, and heuristics that with high probability recover S are designed.
Abstract: We study a semi-random graph model for finding independent sets. For /spl alpha/>0, an n-vertex graph with an independent set S of site /spl alpha/n is constructed by blending random and adversarial decisions. Randomly and independently with probability p, each pair of vertices, such that one is in S and the other is not, is connected by an edge. An adversary can then add edges arbitrarily (provided that S remains an independent set). The smaller p is, the larger the control the adversary has over the semi-random graph. We design heuristics that with high probability recover S when p>(1+/spl epsiv/)ln n/|S|, for any constant /spl epsiv/>0. We show that when p<(1-/spl epsiv/) In n/|S|, an independent set of size |S| cannot be recovered, unless NP/spl sube/BPP. We use our remits to obtain greatly improved coloring algorithms for the model of k-colorable semi-random graphs introduced by A. Blum and J. Spencer (1995).
TL;DR: This work shows that the k-fold clique transversal problem and the maximum h-colourable subgraph problem are polynomially solvable on balanced graphs and provides a polynomial algorithm to recognize balanced graphs.
TL;DR: It is determined that the broadcast times of double and triple fixed step graphs of diameter D are equal to D+2 and D+3, respectively, and it is shown that the diameter of the surviving route graph remains two for any set F of faults for |F|=5, which is optimum.
Abstract: The network properties of double and triple fixed step graphs are considered. We determine that the broadcast times of double and triple fixed step graphs of diameter D are equal to D+2 and D+3, respectively. Some results on the embeddings of grids into these graphs with dilation 1 and 2 are given. For a triple fixed step graph we give a method to calculate the routing between any two vertices of the graph. Furthermore, we show that the diameter of the surviving route graph remains two for any set F of faults for |F|=5, which is optimum.
TL;DR: It is argued that the toughness of split graphs can be computed in polynomial time and this solves an open problem from a recent paper by Kratsch et al.
TL;DR: A simple characterization of graphs which are simultaneouly split and indifference graphs, and a method for optimally edge colouring a complete graph with an even number ⩾6 of vertices, leading to a simple construction for exhibiting a perfect matching of it.
TL;DR: The family of minimal graphs which are clique -complete but have no universal vertices is described, and it is shown that recognizing clique-complete graphs is Co-NP-complete.
TL;DR: The algorithm works for all types of graphs except for a class of highly ambiguous graphs which includes strongly regular graphs, which leaves the option of switching to a higher complexity algorithm if desired.
Abstract: We present an algorithm to solve the graph isomorphism problem for the purpose of object recognition. Objects, such as those which exist in a robot workspace, may be represented by labelled graphs (graphs with attributes on their nodes and/or edges). Thereafter, object recognition is achieved by matching pairs of these graphs. Assuming that all objects are sufficiently different so that their corresponding representative graphs are distinct, then given a new graph, the algorthm efficiently finds the isomorphic stored graph (if it exists). The algorithm consists of three phases: preprocessing, link construction, and ambiguity resolution. Results from experiments on a wide variety and sizes of graphs are reported. Results are also reported for experiments on recognising graphs that represent protein molecules. The algorithm works for all types of graphs except for a class of highly ambiguous graphs which includes strongly regular graphs. However, members of this class are detected in polynomial time, which leaves the option of switching to a higher complexity algorithm if desired.
TL;DR: The main advantage of the new algorithm is that error-correcting graph isomorphism detection is guaranteed to require time that is only polynomial in terms of the size of the input graph.
Abstract: In this paper we present a fast algorithm for the computation of error-correcting graph isomorphisms. The new algorithm is an extension of a method for exact subgraph isomorphism detection from an input graph to a set of a priori known model graphs, which was previously developed by the authors. Similar to the original algorithm, the new method is based on the idea of creating a decision tree from the model graphs. This decision tree is compiled off-line in a preprocessing step. At run time, it is used to find all error-correcting graph isomorphisms from an input graph to any of the model graphs up to a certain degree of distortion. The main advantage of the new algorithm is that error-correcting graph isomorphism detection is guaranteed to require time that is only polynomial in terms of the size of the input graph. Furthermore, the time complexity is completely independent of the number of model graphs and the number of edges in each model graph. However, the size of the decision tree is exponential in the size of the model graphs and the degree of error. Nevertheless, practical experiments have indicated that the method can be applied to graphs containing up to 16 vertices.
TL;DR: The main result implies that overlap graphs have “good” cut-covers, answering an open question of Kaklamanis, Krizanc and Rao (1993), and yields a combinatorial proof that overlap graph have separators of sublinear size.
Abstract: As a special case of our main result, we show that for all L > 0, each k-nearest neighbor graph in d dimensions excludes Kh as a depth L minor if h = Ω(Ld). More generally, we prove that the overlap graphs defined by Miller, Teng, Thurston and Vavasis (1993) have this combinatorial property. By a construction of Plotkin, Rao and Smith (1994), our result implies that overlap graphs have “good” cut-covers, answering an open question of Kaklamanis, Krizanc and Rao (1993). Consequently, overlap graphs can be emulated on hypercube graphs with a constant factor of slow-down and on butterfly graphs with a factor of O(log∗ n) slow-down. Therefore, computations on overlap graphs, such as finite element and finite difference methods on “well-conditioned” meshes and image processing on k-nearest neighbor graphs, can be performed on hypercubic parallel machines with a linear speed-up. Our result, in conjunction with a result of Plotkin, Rao and Smith, also yields a combinatorial proof that overlap graphs have separators of sublinear size. We also show that with high probability, the Delaunay diagram, the relative neighborhood graph, and the k-nearest neighbor graph of a random point set exclude Kh as a depth L minor if h = Ω(Ld2 log n).
TL;DR: In this paper, the local density of triangle-free graphs is estimated based on the spectrum of the graph and on a fractional viewpoint, which is used to refute a conjecture of Erdos et al.
18 Jun 1998
TL;DR: This paper examines the diameter problem on chordal and AT-free graphs and shows that a very simple (linear time) 2-sweep Lex-BFS algorithm identifies a vertex of maximum eccentricity unless the given graph has a specified induced subgraph.
Abstract: Determining the diameter of a graph is a fundamental graph operation, yet no efficient (i.e. quadratic time) algorithm is known. In this paper, we examine the diameter problem on chordal and AT-free graphs and show that a very simple (linear time) 2-sweep Lex-BFS algorithm identifies a vertex of maximum eccentricity unless the given graph has a specified induced subgraph (it was previously known that a single Lex-BFS algorithm is guaranteed to end at a vertex that is within 1 of the diameter for chordal and AT-free graphs). As a consequence of the forbidden induced subgraph result on chordal graphs, our algorithm is guaranteed to work optimally for directed path graphs (it was previously known that a single LexBFS algorithm is guaranteed to work optimally for interval graphs).
TL;DR: A polynomial-time algorithm for approximating the size of the maximumindependent set of an arbitrary graph is described and the computational experiments carried out on 36 DIMACS clique benchmark instances are reported.
Abstract: A family of quadratic programming problems whose optimal values are upper boundson the independence number of a graph is introduced. Among this family, the quadraticprogramming problem which gives the best upper bound is identified. Also the proof thatthe upper bound introduced by Hoffman and Lovasz for regular graphs is a particular caseof this family is given. In addition, some new results characterizing the class of graphs forwhich the independence number attains the optimal value of the above best upper bound aregiven. Finally a polynomial-time algorithm for approximating the size of the maximumindependent set of an arbitrary graph is described and the computational experiments carriedout on 36 DIMACS clique benchmark instances are reported.