Parallel algorithms for P4-comparability graphs

In his pioneering paper on matroids in 1935, Whitney obtained a characterization for binary matroids and left a comment at end of the paper that the problem of characterizing graphic matroids is the same as that of characterizing matroids which correspond to matrices (mod 2) with exactly two ones in each column. Later on Tutte obtained a characterization of graphic matroids in terms of forbidden minors in 1959. It is clear that Whitney indicated about incidence matrices of simple undirected graphs. Here we introduce the concept of a segment binary matroid which corresponds to matrices over $\mathbb{Z}_2$ which has the consecutive $1$'s property (i.e., $1$'s are consecutive) for columns and obtained a characterization of graphic matroids in terms of this. In fact, we introduce a new representation of simple undirected graphs in terms of some vectors of finite dimensional vector spaces over $\mathbb{Z}_2$ which satisfy consecutive $1$'s property. The set of such vectors is called a coding sequence of a graph $G$. Among all such coding sequences we identify the one which is unique for a class of isomorphic graphs. We call it the code of the graph. We characterize several classes of graphs in terms of coding sequences. It is shown that a graph $G$ with $n$ vertices is a tree if and only if any coding sequence of $G$ is a basis of the vector space $\mathbb{Z}_2^{n-1}$ over $\mathbb{Z}_2$. Moreover considering coding sequences as binary matroids, we obtain a characterization for simple graphic matroids and found a necessary and sufficient condition for graph isomorphism in terms of a special matroid isomorphism between their corresponding coding sequences. For this, we introduce the concept of strong isomorphisms of segment binary matroids and show that two simple (undirected) graphs are isomorphic if and only if their canonical sequences are strongly isomorphic segment binary matroids.

In this paper we study the question: How useful is randomization in speeding up Exclusive Write PRAM computations? Our results give further evidence that randomization is of limited use in these types of computations. First we examine a compaction problem on both the CREW and EREW PRAM models, and we present randomized lower bounds which match the best deterministic lower bounds known. (For the CREW PRAM model, the lower bound is asymptotically optimal.) These are the first non-trivial randomized lower bounds known for the compaction problem on these models. We show that our lower bounds also apply to the problem of approximate compaction. Next we examine the problem of computing boolean functions on the CREW PRAM model, and we present a randomized lower bound, which improves on the previous best randomized lower bound for many boolean functions, including the OR function. (The previous lower bounds for these functions were asymptotically optimal, but we improve the constant multiplicative factor.) We also give an alternate proof for the randomized lower bound on PARITY, which was already optimal to within a constant additive factor. Lastly, we give a randomized lower bound for integer merging on an EREW PRAM which matches the best deterministic lower bound known. In all our proofs, we use the Random Adversary method, which has previously only been used for proving lower bounds on models with Concurrent Write capabilities. Thus this paper also serves to illustrate the power and generality of this method for proving parallel randomized lower bounds.

Efficient parallel algorithms are presented, on the CREW PRAM model, for generating a succinct encoding of all pairs shortest path information in a directed planar graphG with real-valued edge costs but no negative cycles. We assume that a planar embedding ofG is given, together with a set ofq faces that cover all the vertices. Then our algorithm runs inO(log2 n) time and employsO(nq+M(q)) processors (whereM(t) is the number of processors required to multiply twot×t matrices inO(logt) time). Let us note here that wheneverq<n then our processor bound is better than the best previous one (M(n)).O(log2 n) time,n-processor algorithms are presented for various subproblems, including that of generating all pairs shortest path information in a directedouterplanar graph. Our work is based on the fundamental hammock-decomposition technique of G. Frederickson. We achieve this decomposition inO(logn log*n) parallel time by usingO(n) processors. The hammock-decomposition seems to be a fundamental operation that may help in improving efficiency of many parallel (and sequential) graph algorithms.

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. Let p be a prime. In this paper, a class of semisymmetric graphs of order 2 p 3 are determined. This work is a partial result for our long term goal to classify all semisymmetric graphs of order 2 p 3 .

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive, but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two biparts of equal size. It was proved by Folkman in (J Comb Theory Ser B 3:215–232, 1967) that there exist no semisymmetric graphs of order $$2p$$ and $$2p^2$$ , where $$p$$ is a prime. For any distinct primes $$p$$ and $$q$$ , the classification of semisymmetric graphs of order $$2pq$$ was given by Du and Xu in (Comm Algebra 28:2685–2715, 2000). Naturally, one of our long-term goals is to determine all the semisymmetric graphs of order $$2p^3$$ , for any prime $$p$$ . All these graphs $$\Gamma $$ are divided into two subclasses: (I) the automorphism group $$\hbox {Aut }(\Gamma )$$ acts unfaithfully on at least one bipart; and (II) $$\hbox {Aut }(\Gamma )$$ acts faithfully on both biparts. In Wang and Du (Eur J Comb 36:393–405, 2014), a group theoretical characterization for Subclass (I) was given by the authors. Based on this characterization, this paper gives a complete classification for Subclass (I).

Given a simple undirected graph G = (V, E) and an integer k < |V|, the Sparsest k-Subgraph problem asks for a set of k vertices which induces the minimum number of edges. As a generalization of the classical independent set problem, Sparsest k-Subgraph is ????-hard and even not approximable unless ?????? in general graphs. Thus, we investigate Sparsest k-Subgraph in graph classes where independent set is polynomial-time solvable, such as subclasses of perfect graphs. Our two main results are the ????-hardness of Sparsest k-Subgraph on chordal graphs, and a greedy 2-approximation algorithm. Finally, we also show how to derive a PTAS for Sparsest k-Subgraph on proper interval graphs.

In this paper we propose a parallel algorithm to construct a one-sided monotone polygon from a Hamiltonian 2-sep chordal graph which takes O(log n) time and uses O(n) processors on a CREW PRAM model. We also propose parallel algorithms to recognize Hamiltonian 2-sep chordal graphs and to construct a Hamiltonian cycle in such a graph, which run in O(log/sup 2/ n) time using O(mn) processors on a CRCW PRAM model and O(log/sup 2/ n) time using O(m) processors on a CREW PRAM model, respectively.

A simple undirected regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. For a semisymmetric graph [Formula: see text] of order [Formula: see text], [Formula: see text] a prime, it is well known that [Formula: see text] is bipartite with two biparts having equal size. The complete classification of such graphs has been given for the full automorphism group [Formula: see text] acting unfaithfully on at least one bipart of [Formula: see text], which shows that there is only one infinite family of such graphs with valency [Formula: see text]. The graphs of this kind have been determined when [Formula: see text] acts faithfully and primitively on at least one bipart of [Formula: see text], and thus there is only one remaining case for classifying such graphs of valency [Formula: see text], [Formula: see text] acting faithfully and imprimitively on both biparts of [Formula: see text], which is dealt with in this paper. As a result, there is only one infinite family of semisymmetric graphs of order [Formula: see text] with valency [Formula: see text].

Graphical models are used to examine conditional independence among random variables. In this chapter we take graphical models for the multivariate complex normal distribution w.r.t. simple undirected graphs into consideration. This is the first published presentation of these models. Graphical models for the multivariate real normal distribution, also called covariance selection models, have already been studied in the literature. The initial work on covariance selection models is done by Dempster (1972) and Wermuth (1976) and a presentation of these models is given in Eriksen (1992). Graphical models for contingency tables are introduced in statistics by Darroch et al. (1980) and further these are well-described in Lauritzen (1989). Graphical association models are treated in general in Whittaker (1990) and Lauritzen (1993). The complex normal graphical models are quite similar to the covariance selection models. We have chosen to develop this chapter without use of exponential families. We study definition of the model, maximum likelihood estimation and hypothesis testing. To verify some of the results in the chapter we use results from mathematical analysis. These can be found in e.g. Rudin (1987). In graphical models one uses the concentration matrix instead of the variance matrix as it is more advantageous. Therefore we define this matrix and derive a relation which is basic for complex normal graphical models. Afterwards we formally define a complex normal graphical model w.r.t. a simple undirected graph. As these models are used to examine conditional independence of selected pairs of variables given the remaining ones we are mainly interested in inference on the concentration matrix. It is possible to base the maximum likelihood estimation of the concentration matrix on a complex random matrix with mean zero.

We introduce here a generic problem component and algorithms for it on various parallel models, that captures the most common, difficult of many types of problems, e.g. geometric dominance. This kernel involves general prefix computations (GPCs) that can, with insight, be done quickly in parallel and by iterative techniques. GPCs' lower bound complexity of Ω (n log n) time is established, and we give optimal solutions on the sequential model in0(n log n) time, on the CREW PRAM model in 0(log n) time with n processors, on the BSR (broadcasting with selective reduction) model in constant time using n processors, and on mesh-connected computers in 0(✓n) time with (✓n) processors. A solution in 0(log2n) time on the hypercube model is given which is the best possible given known sorting limitations. We show that general prefix techniques can be applied to a wide variety of geometric (point set and tree) problems, including triangulation of point sets, two-set dominance counting, ECDF searching, finding two- and three-dimensional maximal points, and the (classical) reconstruction of trees from their traversals. In sum, GPC techniques have many important consequences.

Let [Formula: see text] be a ring with unity. The cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertices are the set of all nonzero and non-unit elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. The reduced cozero-divisor graph of a ring [Formula: see text] is an undirected simple graph whose vertex set is the set of all nontrivial principal ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. Jesili et al. [On the genus of reduced cozero-divisor graph of commutative rings, Soft Comput. 27(2) (2023) 657–666] classified all non-local commutative rings whose reduced cozero-divisor graph is of genus one. In this paper, we characterize all classes of finite non-local commutative rings whose reduced cozero-divisor graph is of genus two. Moreover, we classify all the non-local commutative rings whose cozero-divisor graph is of genus two.

When we want to draw a graph to make the information contained in its structure easily accessible, it is highly desirable to have a drawing with as few edge crossings as possible (Purchase et al., 1997; Purchase, 1997). The class of graphs that can be drawn with no crossings at all is the class of planar graphs. Algorithms for drawing planar graphs are the main subject of this chapter.

Bipartite bithreshold graphs

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The annihilator graph of commutative ring [Formula: see text] is the simple undirected graph AG([Formula: see text]) with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal. In this paper, we classify all finite commutative rings with identity whose annihilator graph and essential graph have crosscap two.

The three-in-a-tree problem is to determine if a simple undirected graph contains an induced subgraph which is a tree connecting three given vertices. Based on a beautiful characterization that is proved in more than twenty pages, Chudnovsky and Seymour [Combinatorica 2010] gave the previously only known polynomial-time algorithm, running in $O(mn^2)$ time, to solve the three-in-a-tree problem on an $n$-vertex $m$-edge graph. Their three-in-a-tree algorithm has become a critical subroutine in several state-of-the-art graph recognition and detection algorithms. In this paper we solve the three-in-a-tree problem in $\tilde{O}(m)$ time, leading to improved algorithms for recognizing perfect graphs and detecting thetas, pyramids, beetles, and odd and even holes. Our result is based on a new and more constructive characterization than that of Chudnovsky and Seymour. Our new characterization is stronger than the original, and our proof implies a new simpler proof for the original characterization. The improved characterization gains the first factor $n$ in speed. The remaining improvement is based on dynamic graph algorithms.