SEPARATION DIMENSION OF BOUNDED DEGREE GRAPHS

The separation dimension of a graph G is the smallest natural number k for which the vertices of G can be embedded in R-k such that any pair of disjoint edges in G can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family F of total orders of the vertices of G such that for any two disjoint edges of G, there exists at least one total order in F in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on n vertices is Theta(log n). In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree d is at most 2(9) (log*d)d. We also demonstrate that the above bound is nearly tight by showing that, for every d, almost all d-regular graphs have separation dimension at least d/2]

Boxicity and cubicity of product graphs

The boxicity (cubicity) of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in R-k. In this article, we give estimates on the boxicity and the cubicity of Cartesian, strong and direct products of graphs in terms of invariants of the component graphs. In particular, we study the growth, as a function of d, of the boxicity and the cubicity of the dth power of a graph with respect to the three products. Among others, we show a surprising result that the boxicity and the cubicity of the dth Cartesian power of any given finite graph is, respectively, in O(log d/ log log d) and circle dot(d/ log d). On the other hand, we show that there cannot exist any sublinear bound on the growth of the boxicity of powers of a general graph with respect to strong and direct products. (C) 2015 Elsevier Ltd. All rights reserved.

Heterochromatic paths in edge colored graphs without small cycles and heterochromatic-triangle-free graphs

Conditions for the existence of heterochromatic Hamiltonian paths and cycles in edge colored graphs are well investigated in literature. A related problem in this domain is to obtain good lower bounds for the length of a maximum heterochromatic path in an edge colored graph G. This problem is also well explored by now and the lower bounds are often specified as functions of the minimum color degree of G - the minimum number of distinct colors occurring at edges incident to any vertex of G - denoted by v(G). Initially, it was conjectured that the lower bound for the length of a maximum heterochromatic path for an edge colored graph G would be 2v(G)/3]. Chen and Li (2005) showed that the length of a maximum heterochromatic path in an edge colored graph G is at least v(G) - 1, if 1 <= v(G) <= 7, and at least 3v(G)/5] + 1 if v(G) >= 8. They conjectured that the tight lower bound would be v(G) - 1 and demonstrated some examples which achieve this bound. An unpublished manuscript from the same authors (Chen, Li) reported to show that if v(G) >= 8, then G contains a heterochromatic path of length at least 120 + 1. In this paper, we give lower bounds for the length of a maximum heterochromatic path in edge colored graphs without small cycles. We show that if G has no four cycles, then it contains a heterochromatic path of length at least v(G) - o(v(G)) and if the girth of G is at least 4 log(2)(v(G)) + 2, then it contains a heterochromatic path of length at least v(G) - 2, which is only one less than the bound conjectured by Chen and Li (2005). Other special cases considered include lower bounds for the length of a maximum heterochromatic path in edge colored bipartite graphs and triangle-free graphs: for triangle-free graphs we obtain a lower bound of 5v(G)/6] and for bipartite graphs we obtain a lower bound of 6v(G)-3/7]. In this paper, it is also shown that if the coloring is such that G has no heterochromatic triangles, then G contains a heterochromatic path of length at least 13v(G)/17)]. This improves the previously known 3v(G)/4] bound obtained by Chen and Li (2011). We also give a relatively shorter and simpler proof showing that any edge colored graph G contains a heterochromatic path of length at least (C) 2015 Elsevier Ltd. All rights reserved.

On the Parameterized Complexity of Finding Separators with Non-Hereditary Properties

We study the problem of finding small s-t separators that induce graphs having certain properties. It is known that finding a minimum clique s-t separator is polynomial-time solvable (Tarjan in Discrete Math. 55:221-232, 1985), while for example the problems of finding a minimum s-t separator that induces a connected graph or forms an independent set are fixed-parameter tractable when parameterized by the size of the separator (Marx et al. in ACM Trans. Algorithms 9(4): 30, 2013). Motivated by these results, we study properties that generalize cliques, independent sets, and connected graphs, and determine the complexity of finding separators satisfying these properties. We investigate these problems also on bounded-degree graphs. Our results are as follows: Finding a minimum c-connected s-t separator is FPT for c=2 and W1]-hard for any ca parts per thousand yen3. Finding a minimum s-t separator with diameter at most d is W1]-hard for any da parts per thousand yen2. Finding a minimum r-regular s-t separator is W1]-hard for any ra parts per thousand yen1. For any decidable graph property, finding a minimum s-t separator with this property is FPT parameterized jointly by the size of the separator and the maximum degree. Finding a connected s-t separator of minimum size does not have a polynomial kernel, even when restricted to graphs of maximum degree at most 3, unless .

On Locally Gabriel Geometric Graphs

Let be a set of points in the plane. A geometric graph on is said to be locally Gabriel if for every edge in , the Euclidean disk with the segment joining and as diameter does not contain any points of that are neighbors of or in . A locally Gabriel graph(LGG) is a generalization of Gabriel graph and is motivated by applications in wireless networks. Unlike a Gabriel graph, there is no unique LGG on a given point set since no edge in a LGG is necessarily included or excluded. Thus the edge set of the graph can be customized to optimize certain network parameters depending on the application. The unit distance graph(UDG), introduced by Erdos, is also a LGG. In this paper, we show the following combinatorial bounds on edge complexity and independent sets of LGG: (i) For any , there exists LGG with edges. This improves upon the previous best bound of . (ii) For various subclasses of convex point sets, we show tight linear bounds on the maximum edge complexity of LGG. (iii) For any LGG on any point set, there exists an independent set of size .

Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

The boxicity (respectively cubicity) of a graph G is the least integer k such that G can be represented as an intersection graph of axis-parallel k-dimensional boxes (respectively k-dimensional unit cubes) and is denoted by box(G) (respectively cub(G)). It was shown by Adiga and Chandran (2010) that for any graph G, cub(G) <= box(G) log(2) alpha(G], where alpha(G) is the maximum size of an independent set in G. In this note we show that cub(G) <= 2 log(2) X (G)] box(G) + X (G) log(2) alpha(G)], where x (G) is the chromatic number of G. This result can provide a much better upper bound than that of Adiga and Chandran for graph classes with bounded chromatic number. For example, for bipartite graphs we obtain cub(G) <= 2(box(G) + log(2) alpha(G)] Moreover, we show that for every positive integer k, there exist graphs with chromatic number k such that for every epsilon > 0, the value given by our upper bound is at most (1 + epsilon) times their cubicity. Thus, our upper bound is almost tight. (c) 2015 Elsevier B.V. All rights reserved.

PolyMage: Automatic Optimization for Image Processing Pipelines

This paper presents the design and implementation of PolyMage, a domain-specific language and compiler for image processing pipelines. An image processing pipeline can be viewed as a graph of interconnected stages which process images successively. Each stage typically performs one of point-wise, stencil, reduction or data-dependent operations on image pixels. Individual stages in a pipeline typically exhibit abundant data parallelism that can be exploited with relative ease. However, the stages also require high memory bandwidth preventing effective utilization of parallelism available on modern architectures. For applications that demand high performance, the traditional options are to use optimized libraries like OpenCV or to optimize manually. While using libraries precludes optimization across library routines, manual optimization accounting for both parallelism and locality is very tedious. The focus of our system, PolyMage, is on automatically generating high-performance implementations of image processing pipelines expressed in a high-level declarative language. Our optimization approach primarily relies on the transformation and code generation capabilities of the polyhedral compiler framework. To the best of our knowledge, this is the first model-driven compiler for image processing pipelines that performs complex fusion, tiling, and storage optimization automatically. Experimental results on a modern multicore system show that the performance achieved by our automatic approach is up to 1.81x better than that achieved through manual tuning in Halide, a state-of-the-art language and compiler for image processing pipelines. For a camera raw image processing pipeline, our performance is comparable to that of a hand-tuned implementation.

Framework for morphometric classification of cells in imaging flow cytometry

Imaging flow cytometry is an emerging technology that combines the statistical power of flow cytometry with spatial and quantitative morphology of digital microscopy. It allows high-throughput imaging of cells with good spatial resolution, while they are in flow. This paper proposes a general framework for the processing/classification of cells imaged using imaging flow cytometer. Each cell is localized by finding an accurate cell contour. Then, features reflecting cell size, circularity and complexity are extracted for the classification using SVM. Unlike the conventional iterative, semi-automatic segmentation algorithms such as active contour, we propose a noniterative, fully automatic graph-based cell localization. In order to evaluate the performance of the proposed framework, we have successfully classified unstained label-free leukaemia cell-lines MOLT, K562 and HL60 from video streams captured using custom fabricated cost-effective microfluidics-based imaging flow cytometer. The proposed system is a significant development in the direction of building a cost-effective cell analysis platform that would facilitate affordable mass screening camps looking cellular morphology for disease diagnosis. Lay description In this article, we propose a novel framework for processing the raw data generated using microfluidics based imaging flow cytometers. Microfluidics microscopy or microfluidics based imaging flow cytometry (mIFC) is a recent microscopy paradigm, that combines the statistical power of flow cytometry with spatial and quantitative morphology of digital microscopy, which allows us imaging cells while they are in flow. In comparison to the conventional slide-based imaging systems, mIFC is a nascent technology enabling high throughput imaging of cells and is yet to take the form of a clinical diagnostic tool. The proposed framework process the raw data generated by the mIFC systems. The framework incorporates several steps: beginning from pre-processing of the raw video frames to enhance the contents of the cell, localising the cell by a novel, fully automatic, non-iterative graph based algorithm, extraction of different quantitative morphological parameters and subsequent classification of cells. In order to evaluate the performance of the proposed framework, we have successfully classified unstained label-free leukaemia cell-lines MOLT, K562 and HL60 from video streams captured using cost-effective microfluidics based imaging flow cytometer. The cell lines of HL60, K562 and MOLT were obtained from ATCC (American Type Culture Collection) and are separately cultured in the lab. Thus, each culture contains cells from its own category alone and thereby provides the ground truth. Each cell is localised by finding a closed cell contour by defining a directed, weighted graph from the Canny edge images of the cell such that the closed contour lies along the shortest weighted path surrounding the centroid of the cell from a starting point on a good curve segment to an immediate endpoint. Once the cell is localised, morphological features reflecting size, shape and complexity of the cells are extracted and used to develop a support vector machine based classification system. We could classify the cell-lines with good accuracy and the results were quite consistent across different cross validation experiments. We hope that imaging flow cytometers equipped with the proposed framework for image processing would enable cost-effective, automated and reliable disease screening in over-loaded facilities, which cannot afford to hire skilled personnel in large numbers. Such platforms would potentially facilitate screening camps in low income group countries; thereby transforming the current health care paradigms by enabling rapid, automated diagnosis for diseases like cancer.