Eulerian Graphs Research Articles

Test Planning in Digital Microfluidic Biochips Using Efficient Eulerization Techniques

Digital microfluidic technology is now being extensively used for implementing a lab-on-a-chip. Microfluidic biochips are often used for safety-critical applications, clinical diagnosis, and for genome analysis. Thus, devising effective and faster testing methodologies to warrant correct operations of these devices after manufacture and during bioassay operations, is very much needed. In this paper, we propose an Euler tour based technique to obtain the route plan of a test droplet for the purpose of structural testing of biochips. The method is applicable to various digital microfluidic biochip architectures, e.g., fully reconfigurable arrays, application specific biochips, pin-constrained irregular geometry biochips, and to defect-tolerant biochips. We show that in general, the optimal Eulerization and subsequent determination of an Euler tour in the graph model of a biochip can be abstracted in terms of the classical Chinese postman problem. The Euler tour can be identified by running the classical Hierholzer's algorithm, which relies on a simple cycle decomposition and splicing method. This improved Eulerization technique leads to an efficient test plan for the chip. This can also be used in phase-based test planning that yields savings in testing time. The method provides a unified approach towards structural testing and can be easily adopted to design a droplet routing procedure for functional testing of digital microfluidic biochips.

Graph theory based segmentation of traced boundary into open and closed sub-sections

Many shape descriptors based on the boundary have been developed for feature extraction and pattern matching/recognition of objects. This paper presents a novel algorithm to segment a traced boundary in open and closed sub-sections. The obtained sub-sections can be utilized for generating more features while being one of the features. A traced boundary has been considered as an undirected graph with pixels representing the vertices. It is proved that “For a boundary traced using 8-connectivity maximum degree of a vertex cannot exceed four”. This aspect and the instances of degree and number of occurrences are used to categorize all the vertices in eight exhaustive types. Six of these are identified as critical vertices. The segmentation is done based on the principle that in an Eulerian circuit, each edge is traversed only once. Further segregation in open and closed sub-graphs is done by choosing critical vertices at a minimum directed distance. The testing of the algorithm has been done on artificially generated images and from different databases of hand written text. The performance analysis shows that the execution time is a linear function of the number of occurrences of each type of the vertices. The performance analysis is further supported using multiple regression.

Edge Extension Of Graphs And Digraphs

In this paper, the concepts of edge (arc) extension of graphs (digraphs) and the edge (arc) extensible class of graphs (digraphs) have been introduced. The classes of regular and eulerian graphs (digraphs) which are not edge (arc) extensible classes have also been introduced. The concept of edge (arc) extensibility number has been introduced as well as the characterization of extensibility number of regular graphs (digraphs). Also the extensibility number of eulerian graphs (digraphs) has been characterized.

Visualization of quasi-median networks

An algorithm for automatic drawing of quasi-median networks is proposed that uses Eulerian cycles in the non-strong-compatibility graph to lay out the network in the plane.

Shuffle Operations on Euler Graphs

The shuffle operation on strings is a fundamental operation, well studied in the theory of formal languages. Shuffle on trajectories yields a flexible method to handle the shuffle operation on two strings. In this paper, the shuffle on trajectories is extended to the string representations of Euler graphs and interesting results are obtained. Some algebraic properties such as completeness, determinism and commutativity of the trajectories involved in this study are provided.

Exact counting of Euler tours for generalized series-parallel graphs

We give a simple polynomial-time algorithm to exactly count the number of Euler tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random ET of the given generalized series-parallel graph. Note that the class of generalized series-parallel graphs includes all outerplanar graphs. We can perform the counting in time O ( m Δ 3 ) , where Δ is the maximum degree of the graph with m edges. We use O ( m n Δ 2 log Δ ) bits to store intermediate values during our computations. To date, these are the first known polynomial-time algorithms to count or sample ETs of any class of graphs; there are no other known polynomial-time algorithms to even approximately count or sample ETs of any other class of graphs. The problem of counting ETs is known to be ♯ P-complete for general graphs (Brightwell and Winkler, 2005 [2]) also for planar graphs (Creed, 2010 [3]).

Parameterized complexity of even/odd subgraph problems

We study the parameterized complexity of the problems of determining whether a graph contains a k-edge subgraph ( k-vertex induced subgraph) that is a Π-graph for Π-graphs being one of the following four classes of graphs: Eulerian graphs, even graphs, odd graphs, and connected odd graphs. We also consider the parameterized complexity of their parametric dual problems. For these sixteen problems, we show that eight of them are fixed parameter tractable and four are W[1]-hard. Our main techniques are the color-coding method of Alon, Yuster and Zwick, and the random separation method of Cai, Chan and Chan.

A new proof of 3-colorability of Eulerian triangulations

Using the existence of noncrossing Eulerian circuits in Eulerian plane graphs, we give a short constructive proof of the theorem of Heawood that Eulerian triangulations are 3-colorable.

Eulerian tour algorithms for data visualization and the PairViz package

PairViz is an R package that produces orderings of statistical objects for visualization purposes. We abstract the ordering problem to one of constructing edge-traversals of (possibly weighted) graphs. PairViz implements various edge traversal algorithms which are based on Eulerian tours and Hamiltonian decompositions. We describe these algorithms, their PairViz implementation and discuss their properties and performance. We illustrate their application to two visualization problems, that of assessing rater agreement, and model comparison in regression.

A Combined Approach for de novo DNA Sequence Assembly of Very Short Reads

De novo DNA sequence assembly is very important in genome sequence analysis. In this paper, we investigated two of the major approaches for de novo DNA sequence assembly of very short reads: overlap-layout-consensus (OLC) and Eulerian path. From that investigation, we developed a new assembly technique by combining the OLC and the Eulerian path methods in a hierarchical process. The contigs yielded by these two approaches were treated as reads and were assembled again to yield longer contigs. We tested our approach using three real very-short-read datasets generated by an Illumina Genome Analyzer and four simulated very-short-read datasets that contained sequencing errors. The sequencing errors were modeled based on Illumina's sequencing technology. As a result, our combined approach yielded longer contigs than those of Edena (OLC) and Velvet (Eulerian path) in various coverage depths and was comparable to SOAPdenovo, in terms of N50 size and maximum contig lengths. The assembly results were also validated by comparing contigs that were produced by assemblers with their reference sequence from an NCBI database. The results show that our approach produces more accurate results than Velvet, Edena, and SOAPdenovo alone. This comparison indicates that our approach is a viable way to assemble very short reads from next generation sequencers.

Using DNA Self-assembly Design Strategies to Motivate Graph Theory Concepts

A number of exciting new laboratory techniques have been developed using the Watson-Crick complementarity properties of DNA strands to achieve the self-assembly of graphical complexes. For all of these methods, an essential step in building the self-assembling nanostructure is designing the component molecular building blocks. These design strategy problems fall naturally into the realm of graph theory. We describe graph theoretical formalism for various construction methods, and then suggest several graph theory exercises to introduce this application into a standard undergraduate graph theory class. This application provides a natural framework for motivating central concepts such as degree sequence, Eulerian graphs, Fleury’s algorithm, trees, graph genus, paths, cycles, etc. There are many open questions associated with these applications which are accessible to students and offer the possibility of exciting undergraduate research experiences in applied graph theory.

The Solution of SAT Problems Using Ternary Vectors and Parallel Processing

This paper will show a new approach to the solution of SAT-problems. It has been based on the isomorphism between the Boolean algebras of finite sets and the Boolean algebras of logic functions depending on a finite number of binary variables. Ternary vectors are the main data structure representing sets of Boolean vectors. The respective set operations (mainly the complement and the intersection) can be executed in a bit-parallel way (64 bits at present), but additionally also on different processors working in parallel. Even a hierarchy of processors, a small set of processor cores of a single CPU, and the huge number of cores of the GPU has been taken into consideration. There is no need for any search algorithms. The approach always finds all solutions of the problem without consideration of special cases (such us no solution, one solution, all solutions). It also allows to include problem-relevant knowledge into the problem-solving process at an early point of time. Very often it is possible to use ternary vectors directly for the modeling of a problem. Some examples are used to illustrate the efficiency of this approach (Sudoku, Queen’s problems on the chessboard, node bases in graphs, graph-coloring problems, Hamiltonian and Eulerian paths etc.). Keywords—SAT–solver, ternary vector, parallel processing, XBOOLE

Fast edge searching and fast searching on graphs

Given a graph G = ( V , E ) in which a fugitive hides on vertices or along edges, graph searching problems are usually to find the minimum number of searchers required to capture the fugitive. In this paper, we consider the problem of finding the minimum number of steps to capture the fugitive. We introduce the fast edge searching problem in the edge search model, which is the problem of finding the minimum number of steps (called the fast edge-search time) to capture the fugitive. We establish relations between the fast edge searching and the fast searching that is the problem of finding the minimum number of searchers to capture the fugitive in the fast search model. While the family of graphs whose fast search number is at most k is not minor-closed for any positive integer k ≥ 2 , we show that the family of graphs whose fast edge-search time is at most k is minor-closed. We establish relations between the fast (fast edge) searching and the node searching. These relations allow us to transform the problem of computing node search numbers to the problem of computing fast edge-search numbers or fast search numbers. Using these relations, we prove that the problem of deciding, given a graph G and an integer k , whether the fast (edge-)search number of G is less than or equal to k is NP-complete; and it remains NP-complete for Eulerian graphs. We also prove that the problem of determining whether the fast (edge-)search number of G is half of the number of odd vertices in G is NP-complete; and it remains NP-complete for planar graphs with maximum degree 4. We present a linear time approximation algorithm for the fast edge-search time that always delivers solutions of at most ( 1 + | V | − 1 | E | + 1 ) times the optimal value. This algorithm also gives us a tight upper bound on the fast search number of graphs. We also show a lower bound on the fast search number using the minimum degree and the number of odd vertices.

The Euler Adic Dynamical System and Path Counts in the Euler Graph

We give a formula for generalized Eulerian numbers, prove monotonicity of sequences of certain ratios of the Eulerian numbers, and apply these results to obtain a new proof that the natural symmetric measure for the Bratteli-Vershik {dynamical} system based on the Euler graph is the unique fully supported invariant ergodic Borel probability measure. Key ingredients of the proof are a two-dimensional induction argument and a one-to-one correspondence between most paths from two vertices at the same level to another vertex.

The Diameter of Almost Eulerian Digraphs

Soares [J. Graph Theory 1992] showed that the well known upper bound $\frac{3}{\delta+1}n+O(1)$ on the diameter of undirected graphs of order $n$ and minimum degree $\delta$ also holds for digraphs, provided they are eulerian. In this paper we investigate if similar bounds can be given for digraphs that are, in some sense, close to being eulerian. In particular we show that a directed graph of order $n$ and minimum degree $\delta$ whose arc set can be partitioned into $s$ trails, where $s\leq \delta-2$, has diameter at most $3 ( \delta+1 - \frac{s}{3})^{-1}n+O(1)$. If $s$ also divides $\delta-2$, then we show the diameter to be at most $3(\delta+1 - \frac{(\delta-2)s}{3(\delta-2)+s} )^{-1}n+O(1)$. The latter bound is sharp, apart from an additive constant. As a corollary we obtain the sharp upper bound $3( \delta+1 - \frac{\delta-2}{3\delta-5})^{-1} n + O(1)$ on the diameter of digraphs that have an eulerian trail.

On Tours that contain all Edges of a Hypergraph

Let $H$ be a $k$-uniform hypergraph, $k\geqslant 2$. By an Euler tour in $H$ we mean an alternating sequence $v_0,e_1,v_1,e_2,v_2,\ldots,v_{m-1},e_m,v_m=v_0$ of vertices and edges in $H$ such that each edge of $H$ appears in this sequence exactly once and $v_{i-1},v_i\in e_i$, $v_{i-1}\neq v_i$, for every $i=1,2,\ldots,m$. This is an obvious generalization of the graph theoretic concept of an Euler tour. A straightforward necessary condition for existence of an Euler tour in a $k$-uniform hypergraph is $|V_{odd}(H)|\leqslant (k-2)|E(H)|$, where $V_{odd}(H)$ is the set of vertices of odd degrees in $H$ and $E(H)$ is the set of edges in $H$. In this paper we show that this condition is also sufficient for hypergraphs of a broad class of $k$-uniform hypergraphs, that we call strongly connected hypergraphs. This result reduces to the Euler theorem on existence of Euler tours, when $k=2$, i.e. for graphs, and is quite simple to prove for $k>3$. Therefore, we concentrate on the most interesting case of $k=3$. In this case we further consider the problem of existence of an Euler tour in a certain class of $3$-uniform hypergraphs containing the class of strongly connected hypergraphs as a proper subclass. For hypergraphs in this class we give a sufficient condition for existence of an Euler tour and prove intractability (NP-completeness) of the problem in this class in general.

The Complexity of Counting Eulerian Tours in 4-regular Graphs

We investigate the complexity of counting Eulerian tours (#ET) and its variations from two perspectives—the complexity of exact counting and the complexity w.r.t. approximation-preserving reductions (AP-reductions, Dyer et al., Algorithmica 38(3):471–500, 2004). We prove that #ET is #P-complete even for planar 4-regular graphs. A closely related problem is that of counting A-trails (#A-trails) in graphs with rotational embedding schemes (so called maps). Kotzig (Theory of Graphs, Proc. Colloq., Tihany, 1966, pp. 219–230, Academic Press, San Diego, 1968) showed that #A-trails can be computed in polynomial time for 4-regular plane graphs (embedding in the plane is equivalent to giving a rotational embedding scheme). We show that for 4-regular maps the problem is #P-hard. Moreover, we show that from the approximation viewpoint #A-trails in 4-regular maps captures the essence of #ET, that is, we give an AP-reduction from #ET in general graphs to #A-trails in 4-regular maps. The reduction uses a fast mixing result for a card shuffling problem (Wilson, Ann. Appl. Probab. 14(1):274–325, 2004).

Adapting parallel algorithms to the W-Stream model, with applications to graph problems

In this paper we show how parallel algorithms can be turned into efficient streaming algorithms for several classical combinatorial problems in the W-Stream model. In this model, at each pass one input stream is read, one output stream is written, and data items have to be processed using limited space; streams are pipelined in such a way that the output stream produced at pass i is given as input stream at pass i+1. We first introduce a simulation technique that allows turning efficient PRAM algorithms into optimal W-Stream ones, for many classical combinatorial problems, including list ranking and Euler tour of a tree. For other problems, most notably graph problems, however, this technique leads to suboptimal algorithms. To overcome this difficulty we introduce the RelaxedPRAM (RPRAM) computational model, as an intermediate model between PRAM and W-Stream. RPRAM allows every processor to access a non-constant number of memory cells per parallel round, albeit with some restrictions. The RPRAM model, while being more powerful than the PRAM model, can be simulated in W-Stream within the same asymptotic bounds. The extra power provided by RPRAM allows us in many cases to substantially reduce the number of processors, while maintaining the same number of parallel rounds, leading to more efficient W-Stream simulations of parallel algorithms. Our RPRAM technique gives new insights on developing streaming algorithms and yields efficient algorithms for several classical problems in this model including sorting, connectivity, minimum spanning tree, biconnected components, and maximal independent set. In addition to allowing smooth space-passes tradeoffs, our algorithms are also shown, by proving almost-tight communication complexity-based lower bounds in W-Stream, to be optimal up to polylog factors.

Image Segmentation using Euler Graphs

This paper presents a new algorithm for image segmentation problem using the concepts of Euler graphs in graph theory. By treating image as an undirected weighted non-planar finite graph (G), image segmentation is handled as graph partitioning problem. The proposed method locates region boundaries or clusters and runs in polynomial time. Subjective comparison and objective evaluation shows the efficacy of the proposed approach in different image domains.

DNA algorithm employing temperature gradient for Freeze-Tag Problem in swarm robotics

The biological deoxyribonucleic acid (DNA) strand is found to be a promising computing unit. An attempt has been made to solve Freeze-Tag Problem (FTP) using DNA. In this paper, the thermodynamic properties of DNA have been utilized along with other biochemical operations to obtain the optimal awakening schedule. Actual distance values are represented using the thermodynamic properties of DNA. The proposed method also finds the minimum spanning tree (MST) of the FTP to obtain the optimal awakening schedule. All possible Euler cycles of the different spanning trees of the problem are first generated. From this generated Euler cycle, the MST is obtained, which is the optimal awakening schedule to the problem. Moreover, the proposed approach can be adopted to solve many real-life applications like broadcasting and scheduling problems, with necessary modifications. In this work, an instance with seven robots is solved using DNA computing.