Preemptive Scheduling of Equal-Length Jobs in Polynomial Time

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

The Median of the Number of Simple Paths on Three Vertices in the Random Graph

We study the asymptotic behavior of the random variable equal to the number of simple paths on three vertices in the binomial random graph in which the edge probability equals the threshold probability of the appearance of such paths. We prove that, for any fixed nonnegative integer b and a sufficiently large number n of vertices of the graph, the probability that the number of simple paths on three vertices in the given random graph is b decreases with n. As a consequence of this result, we obtain the median of the number of simple paths on three vertices for sufficiently large n.

     

Finding a minimum path cover of a distance-hereditary graph in polynomial time

A path cover of a graph G=(V,E) is a set of pairwise vertex-disjoint paths such that the disjoint union of the vertices of these paths equals the vertex set V of G. The path cover problem is, given a graph, to find a path cover having the minimum number of paths. The path cover problem contains the Hamiltonian path problem as a special case since finding a path cover, consisting of a single path, corresponds directly to the Hamiltonian path problem. A graph is a distance-hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them. The complexity of the path cover problem on distance-hereditary graphs has remained unknown. In this paper, we propose the first polynomial-time algorithm, which runs in O(|V|⁹) time, to solve the path cover problem on distance-hereditary graphs.     

On Conditional Covering Problem

The conditional covering problem (CCP) aims to locate facilities on a graph, where the vertex set represents both the demand points and the potential facility locations. The problem has a constraint that each vertex can cover only those vertices that lie within its covering radius and no vertex can cover itself. The objective of the problem is to find a set that minimizes the sum of the facility costs required to cover all the demand points. An algorithm for CCP on paths was presented by Horne and Smith (Networks 46(4):177–185, 2005). We show that their algorithm is wrong and further present a correct O(n ³) algorithm for the same. We also propose an O(n ²) algorithm for the CCP on paths when all vertices are assigned unit costs and further extend this algorithm to interval graphs without an increase in time complexity.     

DNA recombination through assembly graphs

Motivated by DNA rearrangements and DNA homologous recombination modeled in [A. Angeleska, N. Jonoska, M. Saito, L.F. Landweber, RNA-guided DNA assembly, Journal of Theoretical Biology, 248(4) (2007), 706–720], we investigate smoothings on graphs that consist of only 4-valent and 1-valent rigid vertices, called assembly graphs. An assembly graph can be seen as a representation of the DNA during certain recombination processes in which 4-valent vertices correspond to the alignment of the recombination sites. A single gene is modeled by a polygonal path in an assembly graph. A polygonal path makes a “right-angle” turn at every vertex, defining smoothings at the 4-valent vertices and therefore modeling the recombination process. We investigate the minimal number of polygonal paths visiting all vertices of a given graph exactly once, and show that for every positive integer n there are graphs that require at least n such polygonal paths. We show that there is an embedding in three-dimensional space of each assembly graph such that smoothing of vertices according to a given set of polygonal paths results in an unlinked graph. As some recombination processes may happen simultaneously, we characterize the subsets of vertices whose simultaneous smoothings keep a given gene in tact and give a characterization of all sequences of sets of vertices defining successive simultaneous smoothings that can realize complete gene rearrangement.     

Induced Circuits in Planar Graphs

In McDiarmid, B. Reed, A. Schrijver, and B. Shepherd (Univ. of Waterloo Tech. Rep., 1990) a polynomial-time algorithm is given for the problem of finding a minimum cost circuit without chords (induced circuit) traversing two given vertices of a planar graph. The algorithm is based on the ellipsoid method. Here we give an O(n²) combinatorial algorithm to determine whether two nodes in a planar graph lie on an induced circuit. We also give a min-max relation for the problem of finding a maximum number of paths connecting two given vertices in a planar graph so that each pair of these paths forms an induced circuit.     

网络最短的最优解集结构

The shortest path problem in a network G is to find shortest paths between some specified source vertices and terminal vertices when the lengths of edges are given. The structure of the optimal solutions set on the shortest paths is studied in this paper. First,the conditions of having unique shortest path between two distinguished vertices s and t in a network G are discussed;Second,the structural properties of 2-transformation graph (?) on the shortest-paths for G are presented heavily.     

An exact algorithm for subgraph homeomorphism

The subgraph homeomorphism problem is to decide if there is an injective mapping of the vertices of a pattern graph into vertices of a host graph so that the edges of the pattern graph can be mapped into (internally) vertex-disjoint paths in the host graph. The restriction of subgraph homeomorphism where an injective mapping of the vertices of the pattern graph into vertices of the host graph is already given in the input instance is termed fixed-vertex subgraph homeomorphism.We show that fixed-vertex subgraph homeomorphism for a pattern graph on p vertices and a host graph on n vertices can be solved in time 2^n−pn^O(1) or in time 3^n−pn^O(1) and polynomial space. In effect, we obtain new non-trivial upper bounds on the time complexity of the problem of finding k vertex-disjoint paths and general subgraph homeomorphism.     

Constructing disjoint paths on expander graphs

In a typical parallel or distributed computation model processors are connected by a spars interconnection network. To establish open-line communication between pairs of processors that wish to communicate interactively, a set of disjoint paths has to be constructed on the network. Since communication needs vary in time, paths have to be dynamically constructed and destroyed.We study the complexity of constructing disjoint paths between given pairs of vertices on expander interconnection graphs. These graphs have been shown before to possess desirable properties for other communication tasks.We present a sufficient condition for the existence ofKn ^Q edge-disjoint paths connecting any set ofK pairs of vertices on an expander graph, wheren is the number of vertices and<1 is some constant. We then show that the computational problem of constructing these paths lies in the classes Deterministic-P and Random-P C.Furthermore, we show that the set of paths can be constructed in probabilistic polylog time in the parallel-distributed model of computation, in which then participating processors reside in the nodes of the communication graph and all communication is done through edges of the graph. Thus, the disjoint paths are constructed in the very computation model that uses them.Finally, we show how to apply variants of our parallel algorithms to find sets ofvertex-disjoint paths when certain conditions are satisfied.Supported in part by a Weizmann fellowship and by contract ONR N00014-85-C-0731.     

Approximation hardness of min-max tree covers

We prove the first inapproximability bounds to study approximation hardness for a min-max k-tree cover problem and its variants. The problem is to find a set of k trees to cover vertices of a given graph with metric edge weights, so as to minimize the maximum total edge weight of any of the k trees. Our technique can also be applied to improve inapproximability bounds for min-max problems that use other covering objectives, such as stars, paths, and tours.     

On the Convexity Number of Graphs

A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G. We prove NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G) ≥ k. Furthermore, we identify natural necessary extension properties of graphs of small convexity number and study the interplay between these properties and upper bounds on the convexity number.     

Local Algorithms for the Prime Factorization of Strong Product Graphs

The practical application of graph prime factorization algorithms is limited in practice by unavoidable noise in the data. A first step towards error-tolerant “approximate” prime factorization, is the development of local approaches that cover the graph by factorizable patches and then use this information to derive global factors. We present here a local, quasi-linear algorithm for the prime factorization of “locally unrefined” graphs with respect to the strong product. To this end we introduce the backbone \mathbbB (G)\mathbb{B} (G) for a given graph G and show that the neighborhoods of the backbone vertices provide enough information to determine the global prime factors.     

A polynomial-time algorithm of finding a minimum k-path vertex cover and a maximum k-path packing in some graphs

Optimization Letters - For a graph G and a positive integer k, a subset C of vertices of G is called a k-path vertex cover if C intersects all paths of k vertices in G. The cardinality of a minimum...     

Cycles and paths in edge‐colored graphs with given degrees

Sufficient degree conditions for the existence of properly edge‐colored cycles and paths in edge‐colored graphs, multigraphs and random graphs are investigated. In particular, we prove that an edge‐colored multigraph of order n on at least three colors and with minimum colored degree greater than or equal to ?(n+1)/2? has properly edge‐colored cycles of all possible lengths, including hamiltonian cycles. Longest properly edge‐colored paths and hamiltonian paths between given vertices are considered as well. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 63–86, 2010     

Isometric path numbers of graphs

An isometric path between two vertices in a graph G is a shortest path joining them. The isometric path number of G, denoted by ip(G), is the minimum number of isometric paths needed to cover all vertices of G. In this paper, we determine exact values of isometric path numbers of complete r-partite graphs and Cartesian products of 2 or 3 complete graphs.     

Cubic graphs without cut‐vertices having the same path layer matrix

The path layer matrix (or path degree sequence) of a graph G contains quantitative information about all possible paths in G. The entry (i,j) of this matrix is the number of paths in G having initial vertex i and length j. It is known that there are cubic graphs on 62 vertices having the same path layer matrix (A. A. Dobrynin. J Graph Theory 17 (1993) 1–4). A new upper bound of 36 vertices for the least order of such cubic graphs is established. This bound is realized by cubic graphs without cut‐vertices. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 177–182, 2001     

On betweenness-uniform graphs

The betweenness centrality of a vertex of a graph is the fraction of shortest paths between all pairs of vertices passing through that vertex. In this paper, we study properties and constructions of graphs whose vertices have the same value of betweenness centrality (betweenness-uniform graphs); we show that this property holds for distanceregular graphs (which include strongly regular graphs) and various graphs obtained by graph cloning and local join operation. In addition, we show that, for sufficiently large n, there are superpolynomially many betweenness-uniform graphs on n vertices, and explore the structure of betweenness-uniform graphs having a universal or sub-universal vertex.     

The special routes in plane graphs

Let plane undirected graph G = (V, E) be set of items of all manipulator possible trajectories. The problem is constructing of routes satisfying different restrictions. The restrictions can be classified as local and global. For local restrictions the next edge in a route is defined by the conditions set for current vertex or edge. Otherwise restriction is called global. Examples of local restrictions are straightforward paths; a route in which the next edge is defined by the given cycle order on the set of edges incident the current vertex; a route where some edges should be passed in predefined order. The example of global restriction is problem of constructing the cover with ordered enclosing. The paper presents some ways to formalize restrictions and also information about algorithms for constructing of Eulerian covering for plane graph by the sequence of allowed trails. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The crossing number of <Emphasis Type="Italic">G</Emphasis> ▭ <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> for the graph <Emphasis Type="Italic">G</Emphasis> on six vertices

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. The crossing numbers of G□C _n for some graphs G on five and six vertices and the cycle C _n are also given. In this paper, we extend these results by determining the crossing number of the Cartesian product G □ C _n , where G is a specific graph on six vertices.     

A Structure Theorem for Strong Immersions

A graph H is strongly immersed in G if H is obtained from G by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct vertices of G (branch vertices) and edges of H are mapped to pairwise edge‐disjoint paths in G, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We describe the structure of graphs avoiding a fixed graph as a strong immersion. The theorem roughly states that a graph which excludes a fixed graph as a strong immersion has a tree‐like decomposition into pieces glued together on small edge cuts such that each piece of the decomposition has a path‐like linear decomposition isolating the high degree vertices.