Showing papers on "Split graph published in 1990"

Implementing Discrete Mathematics: Combinatorics And Graph Theory With Mathematica

Abstract: Permutations and Combinations Permutations Permutation Groups Inversions and Inversion Vectors Special Classes of Permutations Combinations Exercises and Research Problems * Partitions, Compositions, and Young Tableaux Partitions Compositions Young Tableaux Exercises and Research Problems * Representing Graphs Data Structures for Graphs Elementary Graph Operations Graph Embeddings Storage Formats Exercises and Research Problems * Generating Graphs Regular Structures Trees Random Graphs Relations and Functional Graphs Exercises and Research Problems * Properties of Graphs Connectivity Graph Isomorphism Cycles in Graphs Partial Orders Graph Coloring Cliques, Vertex Covers, and Independent Sets Exercises and Research Problems * Algorithmic Graph Theory Shortest Paths Minimum Spanning Trees Network Flow Matching Planar Graphs Exercises and Research Problems

Approximating Maximum Independent Sets by Excluding Subgraphs

Abstract: An approximation algorithm for the maximum independent set problem is given, improving the best performance guarantee known to \({\cal O}\)(n/(log n)2). We also obtain the same performance guarantee for graph coloring. The results can be combined into a surprisingly strong simultaneous performance guarantee for the clique and coloring problems.

Optimizing weakly triangulated graphs

Abstract: A graph is weakly triangulated if neither the graph nor its complement contains a chordless cycle with five or more vertices as an induced subgraph. We use a new characterization of weakly triangulated graphs to solve certain optimization problems for these graphs. Specifically, an algorithm which runs inO((n + e)n3) time is presented which solves the maximum clique and minimum colouring problems for weakly triangulated graphs; performing the algorithm on the complement gives a solution to the maximum stable set and minimum clique covering problems. Also, anO((n + e)n4) time algorithm is presented which solves the weighted versions of these problems.

A branch and bound algorithm for the maximum clique problem

Abstract: We present a branch and bound algorithm for the maximum clique problem in arbitrary graphs. The main part of the algorithm consists in the determination of upper bounds by graph colorings. Using a modification of a known graph coloring method called DSATUR we simultaneously derive lower and upper bounds for the clique number.

Exploiting the special structure of conflict and compatibility graphs in high-level synthesis

Abstract: Coloring of conflict graphs and clique partitioning of compatibility graphs have been used in high-level synthesis to map operators, values, and data transfers onto shared resources. However, finding a minimum sized coloring or clique partition is NP hard. One method to overcome this complexity is to identify special types of graphs that can be colored or clique partitioned in polynomial time. Existing high-level synthesis systems have exploited two special types of conflict graphs-interval and circular-arc graphs. However, they have provided no insight into why and how frequently these graphs occur. This paper will investigate the features of behavioral representations and synthesis algorithms that give rise to special conflict and compatibility graphs. We will identify two additional types of graphs useful for high-level synthesis-chordal and comparability graphs-and demonstrate their use in an existing high-level synthesis system. >

Size and independence in triangle-free graphs with maximum degree three

Abstract: Let C be the class of triangle-free graphs with maximum degree at most three. A lower bound for the number of edges in a graph of C is derived in terms of the number of vertices and the independence. Several classes of graphs for which this bound is attained are given. As corollaries, we obtain the best possible lower bound for the independence ratio of a graph in C and evaluate some Ramsey-type numbers.

An approximation algorithm for coloring circular-arc graphs

Abstract: Consider families of arcs on a circle. The minimum coloring problem on arc families has been shown to be NP-hard by Garey, Johnson, Miller and Papadimitriou. It is easy to show that 2q colors are sufficient for any arc family F, where q is the size of a maximum clique in F and 3q/2 colors are necessary for some families. It has long been open problem to find a coloring algorithm which uses no more than α·q colors , where α is strictly less than 2. In this paper we present such an algorithm with α=5/3. Our algorithm is based on: (1) an extension of an earlier result of Tucker on coloring special families and (2) a characterization of the existence of perfect matching in bipartite graphs. 1 Department of Computer Science, National Tsing-Hua University, Republic of China. 2 Institute of Information Sciences, Academia Sinica, Republic of China.

A recognition algorithm for II-graphs

Abstract: This thesis studies an extension of both interval and permutation graphs known as II-graphs (Interval-Interval or trapezoidal graphs). Given two parallel horizontal lines with n intervals on each line, for any interval of the top line there is exactly one trapezoid joining it with an interval of the lower line. Each such trapezoid corresponds to a vertex of the II-graph, where two vertices are adjacent if and only if their corresponding trapezoids intersect. It is known that II-graphs exhibit many interesting properties such as being weakly chordal, co-comparability and asteroidal-triple free. Their complements are transitively orientable with an interval order of dimension two. This thesis presents an $O(n\sp3$) algorithm for solving the II-graph recognition problem. Using an operation for "splitting" a vertex into two (called the Vertex Splitting Operation), our recognition algorithm will transform a given graph into a permutation graph with some special properties if and only if the given graph is an II-graph. Unlike other II-graph recognition algorithms, our algorithm will also construct an II-representation. We will also show that the Vertex Splitting Operation exhibits various interesting properties when applied to other families of graphs, including perfect graphs, chordal graphs and interval graphs.

Diameters of iterated clique graphs of chordal graphs

Abstract: The clique graph K(G) of a graph is the intersection graph of maximal cliques of G. The iterated clique graph Kn(G) is inductively defined as K(Kn−1(G)) and K1(G) = K(G). Let the diameter diam(G) be the greatest distance between all pairs of vertices of G. We show that diam(Kn(G)) = diam(G) — n if G is a connected chordal graph and n ≤ diam(G). This generalizes a similar result for time graphs by Bruce Hedman.

Algorithms on special classes of graphs and partially ordered sets

Abstract: This thesis addresses a variety of problems on special classes of graphs and partially ordered sets. Special classes of graphs arise in many applications. On these classes of graphs, many problems are found to have faster algorithms. We first present a fast algorithm for split decomposition. Split decomposition has been used in solving other problems. Recently, it provides the basis for recognition algorithms on circle graphs. Substitution decomposition is a more restricted version of split decomposition. We introduce a linear time algorithm to perform substitution decomposition on chordal graphs. Some potential applications are also pointed out. Matrix multiplication is a commonly used tool in solving many combinatorial problems. For example, it provides the fastest algorithm to solve the transitive closure and the neighborhood containment problems. We show that by using a characterization of N-free posets, we can avoid matrix multiplication and find a faster algorithm for the transitive closure problem on N-free posets. The chain subgraph cover problem has been a useful vehicle in proving NP-completeness of many open problems. In this thesis, we further explore the relationship between chain graphs and posets. We also find many problems which can be easily transformed into the two chain subgraph cover problem. By presenting an O($n\sp2$) reduction from the two chain subgraph cover problem to the partial order dimension two problem, we develop O($n\sp2$) algorithms for bidimension two, Ferrers dimension two, interval dimension two, and trapezoid graph recognition problems. The dimension of posets is one of the most studied invariant of posets. A poset is called cycle-free if its comparability graph is chordal. Using the clique tree representations of chordal graphs, we prove that every cycle-free poset is at most 4 dimensional.

On the construction of optimally reliable graphs

Abstract: We consider the reliability of regular graphs of degree k in which the vertices of the graph fail independently with probability p. The probability of disconnection is minimized for small values of p if the graph has connectivity k and has the smallest number of vertex cut sets with k vertices. In this paper, we show how to construct such graphs.

A parallel algorithm for finding a maximum weight clique of an interval graph

Abstract: In this paper we study the problem of finding a maximum weight clique of an interval graph. Let n be the number of vertices in the graph. We show that a maximum weight clique of an interval graph can be found in time O(log n) using n processors on an exclusive-read exclusive-write parallel random access machine.

Fast parallel algorithms for the clique separator decomposition

Abstract: We give an efficient {\it NC} algorithm for finding a clique separator decomposition of an {\it arbitrary} graph, that is, a series of cliques whose removal disconnects the graph. This algorithm allows one to extend a large body of results which were originally formulated for chordal graphs to other classes of graphs. Our algorithm is optimal to within a polylogarithmic factor of Tarjan''s $O(mn)$ time sequential algorithm. The decomposition can also be used to find {\it NC} algorithms for some optimization problems on special families of graphs, assuming these problems can be solved in {\it NC} for the prime graphs of the decomposition. These optimization problems include: finding a maximum-weight clique, a minimum coloring, a maximum-weight independent set, and a minimum fill-in elimination order. We also give the first parallel algorithms for solving these problems by using the clique separator decomposition. Our maximum-weight independent set algorithm applied to chordal graphs yields the most efficient known parallel algorithm for finding a maximum-weight independent set of a chordal graph.

Coloring certain proximity graphs

Abstract: We examine the graph coloring problem for three families of Euclidean proximity graphs. Results include a linear-time 4-coloring algorithm for relative neighborhood graphs , a linear-time 3-coloring method for 3-chromatic Delaunay graphs , and two minimum-coloring heuristics for Gabriel graphs . The heuristics are shown to outperform other coloring methods when applied to these graphs. We also show that the 3-colorability problem for Delaunay graphs is polynomial-time solvable.

Parallel algorithms for intersection graphs

Abstract: Intersection graphs have often been used to model special structure in graph problems for both practical and theoretical reasons. Often problems can be solved efficiently for a restricted class of graphs that are provably hard for graphs in general. In this thesis, we examine parallel (i.e. NC) algorithms for exploiting the special structure of different types of intersection graphs to solve common graph problems. First, we show how to recognize whether a graph belongs to one of the special families of intersection graphs. Only after this has been done can we take advantage of the properties of a class of intersection graphs. Let $\cal F$ be a family of nonempty sets. Then the intersection graph of $\cal F$ is obtained by representing each set in $\cal F$ by a vertex and connecting two vertices if and only if their corresponding sets have a nonempty intersection. In this dissertation several types of intersection graphs will be examined, among them interval, comparability, chordal, path, and circle graphs. Interval and comparability graphs arise often when solving scheduling problems. Chordal graphs have applications in solving sparse systems of linear equations and in relational database theory. Each of these classes of intersection graphs is closed under a natural graph composition operation, namely one of modular composition, clique identification, and split composition. In this thesis, we show that the number of intersection representations for a graph in one of these classes depends on how the graphs was composed from indecomposable graphs. Also, we show how to efficiently decompose any graph (i.e. perform the inverse of the above composition operations) in parallel. Often if we can solve a problem efficiently on the indecomposable pieces of a graph, then we can solve this same problem efficiently on the graph itself in parallel. Thus we can extend many of our results on special classes of intersection graphs to more general classes of graphs.

Generalized clique coverings of chordal graphs.

Abstract: Two main types 1)£ clique coveri cd phs ha'J(? d j,scussed. One is ali que c C) v p r i r, 9 Cl f \! l-''', ice S , a set ,-, f clIques which between them contain eve! 'JI? te:,c at once. The other, a clique c:-overing of edgEs, is a set f cliques which between them contain eVPIY e0ge. CllVering of edges may be deflned as a lique covering of vertices with the added rest iction hat tho=> I:'"nds of each edge must together belong to at least one clique. ::::uppose ~ '" {51' 52'"'' 5 k } is a family of sets, where t hI'? el"o?lIlen t s of S i may be vext ices or edges of a graph G. A.

Optimal Parallel Algorithms for Sparse Graphs

Abstract: We present here techniques which exhibit optimal processor-time tradeoff for many important problems on sparse graphs. These problems include: maximal coloring and maximal independent set in trees and bounded degree graphs; 7-colorability, maximal independent set and maximal matching in planar graphs; maximum independent set, maximum matching and Hamiltonian path on rectangular grid graphs. Our techniques are based on the general list ranking problem: given k lists having a total of n elements, find for each element the membership relation and the rank of the element in its list. We solve this problem in O(log n) time with n/log n processors on an EREW PRAM. For trees and bounded degree graphs our methods need O(log n) time and n/log n processors on an EREW PRAM. For planar graphs they need O(log2n) time and n/log2n processors on an EREW PRAM using linear space. For the case of rectangular grid graphs they need O(log n) time with n/log n processors on a CRCW PRAM, or on an EREW PRAM (if the embedding is given).

An nc algorithm to recognize hhd-free graphs

Abstract: The HHD-free graphs are a class of perfect graphs that strictly contain the cographs, the chordal graphs, the Matula-perfect graphs, and the Welsh-Powell opposition graphs. In this paper we present two characterizations of the HHD-free graphs and show how to use them to produce a parallel recognition algorithm which runs in O(log n) time using O(n 4) processors on a concurrent-read, concurrent-write parallel random access machine.

Parallel algorithms for special graphs

Abstract: Some decision problems like independent set, dominating set, clique, and coloring are all known to be NP-complete for a general graph. When, however, we restrict the graph to the special classes of intersection graphs, such as permutation graphs, circle graphs, and circular-arc graphs, many of these problems fall into the P class. These intersection graphs are known to be quite useful in modeling real-world problems arising in computer CPU and memory scheduling, VLSI layout, and so on. As a result, we have a large collection of polynomial time algorithms for many graph problems defined on these graphs. In particular, all the three classes of graphs listed above can be recognized in polynomial time. It is strongly conjectured that not all the problems in P lie in NC and the existence of the so-called P-complete problems supports this argument. Therefore, it should be interesting to reveal that which of the problems in P are also in NC. In this dissertation, we study the permutation graphs, circle graphs, and circular-arc graphs by showing a series of NC algorithms for the problems mentioned above. Our algorithms for the permutation graphs and the circle graphs are new and hence these problems are proved to be in NC for the first time. For the circular-arc graphs, we present optimal EREW parallel algorithms for finding a maximum independent set and a minimum dominating set.