Efficient parallel recognition of some circular arc graphs,I

We present the first efficient parallel algorithms for recognizing some subclasses of circular arc graphs including circular arc graphs and proper interval graphs. These algorithms run in O(log² n) time withO(n ³) processors on a CRCW PRAM. An intersection representation can also be constructed within the same resource bounds. Furthermore, we propose some new characterizations of circular arc graphs and proper interval graphs.Portions of this paper have appeared in preliminary form in theProceedings of the 1989 IEEE international Symposium on Circuits and Systems [9], theProceedings of the 1989 Workshop on Algorithms and Data Structures [10], and theProceedings of the 1990 Canadian Conference on Computational Geometry [11].     

A linear time recognition algorithm for proper interval graphs

We propose a linear time recognition algorithm for proper interval graphs. The algorithm is based on certain ordering of vertices, called bicompatible elimination ordering (BCO). Given a BCO of a biconnected proper interval graph G, we also propose a linear time algorithm to construct a Hamiltonian cycle of G.     

A parallel algorithm for generating bicompatible elimination orderings of proper interval graphs

In this paper, we first show how a certain ordering of vertices, called bicompatible elimination ordering (BCO), of a proper interval graph can be used to solve efficiently several problems in proper interval graphs. We, then, propose an NC parallel algorithm (i.e., polylogarithmic-time employing a polynomial number of processors) in SIMD CRCW PRAM (Single Instruction Stream Multiple Data Stream Concurrent Read Concurrent Write Parallel Random Access Machine) model of parallel computation to compute a BCO of a proper interval graph. To the best of our knowledge, this is the first NC parallel algorithm to compute a BCO of a proper interval graph.     

Efficient algorithms for single- and two-layer linear placement of parallel graphs

This paper outlines an algorithm for optimum linear ordering (OLO) of a weighted parallel graph with O(n log k) worst-case time complexity, and O(n + k log(n/k) log k) expected-case time complexity, where n is the total number of nodes and k is the number of chains in the parallel graph. Next, the two-layer OLO problem is considered, where the goal is to place the nodes linearly in two routing layers minimizing the total wire length. The two-layer problem is shown to subsume the maxcut problem and a befitting heuristic algorithm is proposed. Experimental results on randomly generated samples show that the heuristic algorithm runs very fast and outputs optimum solutions in more than 90% instances.     

Faster recognition of clique-Helly and hereditary clique-Helly graphs

A family of subsets of a set is Helly when every subfamily of it, which is formed by pairwise intersecting subsets contains a common element. A graph G is clique-Helly when the family of its (maximal) cliques is Helly, while G is hereditary clique-Helly when every induced subgraph of it is clique-Helly. The best algorithms currently known to recognize clique-Helly and hereditary clique-Helly graphs have complexities O(nm²) and O(n²m), respectively, for a graph with n vertices and m edges. In this Note, we describe algorithms which recognize both classes in O(m²) time. These algorithms also reduce the complexity of recognizing some other classes, as disk-Helly graphs.     

A note on the Hadwiger number of circular arc graphs

The intention of this note is to motivate the researchers to study Hadwiger's conjecture for circular arc graphs. Let η(G) denote the largest clique minor of a graph G, and let χ(G) denote its chromatic number. Hadwiger's conjecture states that η(G)?χ(G) and is one of the most important and difficult open problems in graph theory. From the point of view of researchers who are sceptical of the validity of the conjecture, it is interesting to study the conjecture for graph classes where η(G) is guaranteed not to grow too fast with respect to χ(G), since such classes of graphs are indeed a reasonable place to look for possible counterexamples. We show that in any circular arc graph G, η(G)?2χ(G)−1, and there is a family with equality. So, it makes sense to study Hadwiger's conjecture for this family.     

A selected tour of the theory of identification matrices

A selected tour of the theory of identification matrices is offered here. We show that, among other things, shortest-path adjacency matrices are identification matrices for all simple graphs and adjacency matrices are identification matrices for all bipartite graphs. Additionally, we provide an improved proof that augmented adjacency matrices satisfying the circular 1's property are identification matrices. We also present a characterization of doubly convex bipartite graphs by identification matrices. Based on the theory of identification matrices, we describe an improved method for testing isomorphism between Γ circular arc graphs. The sequential algorithm can be implemented to run in O(n²) time and is optimal if the graphs are given as (augmented) adjacency matrices, so to speak.     

Locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs

A spanning tree T of a graph G=(V,E) is called a locally connected spanning tree if the set of all neighbors of v in T induces a connected subgraph of G for all v∈V. The problem of recognizing whether a graph admits a locally connected spanning tree is known to be NP-complete even when the input graphs are restricted to chordal graphs. In this paper, we propose linear time algorithms for finding locally connected spanning trees in cographs, complements of bipartite graphs and doubly chordal graphs, respectively.     

An online composite graphics recognition approach based on matching of spatial relation graphs

A spatial relation graph (SRG) and its partial matching method are proposed for online composite graphics representation and recognition. The SRG-based approach emphasizes three characteristics of online graphics recognition: partial, structural, and independent of stroke order and stroke number. A constrained partial permutation strategy is also proposed to reduce the computational cost of matching two SRGs, which is originally an NP-complete problem as is graph isomorphism. Experimental results show that our proposed SRG-based approach is both efficient and effective for online composite graphics recognition in our sketch-based graphics input system - SmartSketchpad.Received: 13 March 2003, Accepted: 13 March 2004, Published online: 1 June 2004     

Structural matching of 2D electrophoresis gels using deformed graphs

2D electrophoresis is a well-known method for protein separation which is extremely useful in the field of proteomics. Each spot in the image represents a protein accumulation and the goal is to perform a differential analysis between pairs of images to study changes in protein content. It is thus necessary to register two images by finding spot correspondences. Although it may seem a simple task, generally, the manual processing of this kind of images is very cumbersome, especially when strong variations between corresponding sets of spots are expected (e.g. strong non-linear deformations and outliers). In order to solve this problem, this paper proposes a new quadratic assignment formulation together with a correspondence estimation algorithm based on graph matching which takes into account the structural information between the detected spots. Each image is represented by a graph and the task is to find a maximum common subgraph. Successful experimental results using real data are presented, including an extensive comparative performance evaluation with ground-truth data.