Coloring permutation graphs in parallel

A coloring of a graph G is an assignment of colors to its vertices so that no two adjacent vertices have the same color. We study the problem of coloring permutation graphs using certain properties of the lattice representation of a permutation and relationships between permutations, directed acyclic graphs and rooted trees having specific key properties. We propose an efficient parallel algorithm which colors an n-node permutation graph in O(log² n) time using O(n²/log n) processors on the CREW PRAM model. Specifically, given a permutation π we construct a tree T^*[π], which we call coloring-permutation tree, using certain combinatorial properties of π. We show that the problem of coloring a permutation graph is equivalent to finding vertex levels in the coloring-permutation tree.     

A new infinite series of regular uniformly geodetic code graphs

We construct vertex-transitive graphs Γ, regular of valency k=n²+n+1 on vertices, with integral spectrum, possessing a distinguished complete matching such that contracting the edges of this matching yields the Johnson graph J(2n, n) (of valency n²). These graphs are uniformly geodetic in the sense of Cook and Pryce (1983) (F-geodetic in the sense of Ceccharini and Sappa (1986)), i.e., the number of geodesics between any two vertices only depends on their distance (and equals 4 when this distance is two). They are counterexamples to Theorem 3.15.1 of [1], and we show that there are no other counterexamples.     

ContributionsAn algorithm for the Tutte polynomials of graphs of bounded treewidth

Let k be a fixed, positive integer. We give an algorithm which computes the Tutte polynomial of any graph G of treewidth at most k in time O(n^{2+7 log₂ c}), where c is twice the number of partitions of a set with 3k + 3 elements and n the number of vertices of G.     

Diagonal flips in pseudo-triangulations on closed surfaces

Negami has already shown that there is a natural number N(F²) for any closed surface F² such that two triangulations on F² with n vertices can be transformed into each other by a sequence of diagonal flips if nN(F²). We investigate the same theorem for pseudo-triangulations with or without loops, estimating the length of a sequence of diagonal flips. Our arguments will be applied to simple triangulations to obtain a linear upper bound for N(F²) with respect to the genus of F².     

Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

The Ramsey number r(H,K_n) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K_2,m,K_n)(m−1+o(1))(n/log n)² and r(C_2m,K_n)c(n/log n)^m/(m−1) for m fixed and n→∞. Also r(K_2,n,K_n)=Θ(n³/log² n) and .     

The coordinate representation of a graph and n-universal graph of radius 1

Every graph can be represented as the intersection graph on a family of closed unit cubes in Euclidean space Eⁿ. Cube vertices have integer coordinates. The coordinate matrix, A(G)={v_nk} of a graph G is defined by the set of cube coordinates. The imbedded dimension of a graph, Bp(G), is a number of columns in matrix A(G) such that each of them has at least two distinct elements v_nk≠v_pk. We show that Bp(G)=cub(G) for some graphs, and Bp(G)n−2 for any graph G on n vertices. The coordinate matrix uses to obtain the graph U of radius 1 with 3ⁿ⁻² vertices that contains as an induced subgraph a copy of any graph on n vertices.     

The Second Neighbourhood for Quasi-transitive Oriented Graphs

In 2006, Sullivan stated the conjectures:(1) every oriented graph has a vertex x such that d~(++)(x) ≥ d~-(x);(2) every oriented graph has a vertex x such that d~(++)(x) + d~+(x) ≥ 2 d~-(x);(3) every oriented graph has a vertex x such that d~(++)(x) + d~+(x) ≥ 2 · min{d~+(x), d~-(x)}. A vertex x in D satisfying Conjecture(i) is called a Sullivan-i vertex, i = 1, 2, 3. A digraph D is called quasi-transitive if for every pair xy, yz of arcs between distinct vertices x, y, z, xz or zx("or" is inclusive here) is in D. In this paper, we prove that the conjectures hold for quasi-transitive oriented graphs, which is a superclass of tournaments and transitive acyclic digraphs. Furthermore, we show that a quasi-transitive oriented graph with no vertex of in-degree zero has at least three Sullivan-1 vertices and a quasi-transitive oriented graph has at least three Sullivan-3 vertices unless it belongs to an exceptional class of quasitransitive oriented graphs. For Sullivan-2 vertices, we show that an extended tournament, a subclass of quasi-transitive oriented graphs and a superclass of tournaments, has at least two Sullivan-2 vertices unless it belongs to an exceptional class of extended tournaments.     

On-line construction of the upper envelope of triangles and surface patches in three dimensions

In this paper, we describe a randomized incremental algorithm for computing the upper envelope (i.e., the pointwise maximum) of a set of n triangles in three dimensions. This algorithm is an on-line algorithm. It is structure-sensitive: the expected cost of inserting the n-th triangle is O(log nΣ_r=1ⁿτ(r)/r²) and depends on the expected size τ(r) of an intermediate result for r triangles. Since τ(r) can be Θ(r²(r)) in the worst case, this cost is bounded in the worst case by O(n(n) log n). (The expected behaviour is analyzed by averaging over all possible orderings of the input.) The main new characteristics is the use of a two-level history graph. (The history graph is an auxiliary data structure maintained by randomized incremental algorithms.) Our algorithm is fairly simple and appears to be efficient in practice. It extends to surfaces and surface patches of fixed maximum algebraic degree.     

Dominating the complements of bounded tolerance graphs and the complements of trapezoid graphs

We investigate the complexity of several domination problems on the complements of bounded tolerance graphs and the complements of trapezoid graphs. We describe an O(n² log⁵ n) time and O(n²) space algorithm to solve the domination problem on the complement of a bounded tolerance graph, given a square embedding of that graph. We also prove that domination, connected domination and total domination are all NP-complete on co-trapezoid graphs.     

On the average complexity of 3D-Voronoi diagrams of random points on convex polytopes

It is well known that the complexity, i.e. the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as Θ(n²). It is also known that if the points are chosen Independently Identically Distributed uniformly from a 3-dimensional region such as a cube or sphere, then the expected complexity falls to O(n). In this paper we introduce the problem of analyzing what occurs if the points are chosen from a 2-dimensional region in 3-dimensional space. As an example, we examine the situation when the points are drawn from a Poisson distribution with rate n on the surface of a convex polytope. We prove that, in this case, the expected complexity of the resulting Voronoi diagram is O(n).     

Unavoidable doubly connected large graphs

A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h(n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs:

(1) P_n, a path on n vertices;

(2) K_1,n^s, the graph obtained from K_1,n by subdividing an edge once;

(3) K_2,ne, the graph obtained from K_2,n by deleting an edge;

(4) K_2,n⁺, the graph obtained from K_2,n by adding an edge between the two degree-n vertices x₁ and x₂, and a pendent edge at each x_i.

Two applications of this result are also discussed in the paper.     

A note on G-intersecting families

Consider a graph G and a k-uniform hypergraph on common vertex set [n]. We say that is G-intersecting if for every pair of edges in there are vertices xX and yY such that x=y or x and y are joined by an edge in G. This notion was introduced by Bohman, Frieze, Ruszinkó and Thoma who proved a natural generalization of the Erd s–Ko–Rado Theorem for G-intersecting k-uniform hypergraphs for G sparse and k=O(n^1/4). In this note, we extend this result to .     

Optimal fault-tolerant routings with small routing tables for k-connected graphs

We study the problem of designing fault-tolerant routings with small routing tables for a k-connected network of n processors in the surviving route graph model. The surviving route graph R(G,ρ)/F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D(R(G,ρ)/F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n2k², in which the route degree is , the total number of routes is O(k²n) and D(R(G,λ)/F)3 for any fault set F (|F|<k). In particular, in the case that k=2 we can construct a routing λ′ for every biconnected graph in which the route degree is , the total number of routes is O(n) and D(R(G,λ′)/{f})3 for any fault f. We also show that we can construct a routing ρ₁ for every n-node biconnected graph, in which the total number of routes is O(n) and D(R(G,ρ₁)/{f})2 for any fault f, and a routing ρ₂ (using ρ₁) for every n-node biconnected graph, in which the route degree is , the total number of routes is and D(R(G,ρ₂)/{f})2 for any fault f.     

Efficient visibility queries in simple polygons

We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n³logn) preprocessing time and O(n³) space, we can, given a query point q inside or outside an n vertex polygon, recover the visibility polygon of q in O(logn+k) time, where k is the size of the visibility polygon, and recover the number of vertices visible from q in O(logn) time.

The key notion behind the decomposition is the succinct representation of visibility regions, and tight bounds on the number of such regions. These techniques are extended to handle other types of queries, such as visibility of fixed points other than the polygon vertices, and for visibility from a line segment rather than a point. Some of these results have been obtained independently by Guibas, Motwani and Raghavan [18] .     

Computing rectangle enclosures

The rectangle enclosure problem is the problem of determining the subset of n iso-oriented planar rectangles that enclose a query rectangle Q. In this paper, we use a three layered data structure which is a combination of Range and Priority search trees and answers both the static and dynamic cases of the problem. Both the cases use O(n> log² n) space. For the static case, the query time is O(log² n log log n + K). The dynamic case is supported in O(log³ n + K) query time using O(log³ n) amortized time per update. K denotes the size of the answer. For the d-dimensional space the results are analogous. The query time is O(log^2d-2 n log log n + K) for the static case and O(log^2d-1 n + K) for the dynamic case. The space used is O(n> log^2d-2 n) and the amortized time for an update is O(log^2d-1 n). The existing bounds given for a class of problems which includes the present one, are O(log^2d n + K) query time, O(log^2d n) time for an insertion and O(log^2d-1 n) time for a deletion.     

Popular distances in 3-space

Let m(n) denote the smallest integer m with the property that any set of n points in Euclidean 3-space has an element such that at most m other elements are equidistant from it. We have that cn^1/3 log log n m(n) n^3/5 β(n), where c> 0 is a constant and β(n) is an extremely slowly growing function, related to the inverse of the Ackermann function.     

Computing simple paths among obstacles

Given a set X of points in the plane, two distinguished points s,tX, and a set Φ of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in Φ. We present two results: (1) we show that finding such simple paths among arbitrary obstacles is NP-complete, and (2) we give a polynomial-time algorithm that computes simple paths when the obstacles form a simple polygon P and X is inside P. Our algorithm runs in time O(m²n²), where m is the number of vertices of P and n is the number of points in X.     

q-distributions and Markov processes

We consider a sequence of integer-valued random variables X_n, n 1, representing a special Markov process with transition probability λ_{n, l}, satisfying P_{n, l} = (1 − λ_{n, l}) P_{n−1, l} + λ_{n, l−1} P_{n−1, l−1}. Whenever the transition probability is given by λ_{n, l} = q^{n + βl + γ} and λ_{n, l} = 1 − q^n+βl+γ, we can find closed forms for the distribution and the moments of the corresponding random variables, showing that they involve functions such as the q-binomial coefficients and the q-Stirling numbers. In general, it turns out that the q-notation, up to now mainly used in the theory of q-hypergeometrical series, represents a powerful tool to deal with these kinds of problems. In this context we speak therefore about q-distributions. Finally, we present some possible, mainly graph theoretical interpretations of these random variables for special choices of , β and γ.     

Consequences of an algorithm for bridged graphs

Chepoi showed that every breadth first search of a bridged graph produces a cop-win ordering of the graph. We note here that Chepoi's proof gives a simple proof of the theorem that G is bridged if and only if G is cop-win and has no induced cycle of length four or five, and that this characterization together with Chepoi's proof reduces the time complexity of bridged graph recognition. Specifically, we show that bridged graph recognition is equivalent to (C₄,C₅)-free graph recognition, and reduce the best known time complexity from O(n⁴) to O(n^3.376).     

Dynamic motion planning in low obstacle density environments

A fundamental task for an autonomous robot is to plan its own motions. Exact approaches to the solution of this motion planning problem suffer from high worst-case running times. The weak and realistic low obstacle density (L.O.D.) assumption results in linear complexity in the number of obstacles of the free space (Van der Stappen et al., 1997). In this paper we address the dynamic version of the motion planning problem in which a robot moves among polygonal obstacles which move along polylines. The obstacles are assumed to move along constant complexity polylines, and to respect the low density property at any given time. We will show that in this situation a cell decomposition of the free space of size O(n²(n) log² n) can be computed in O(n²(n) log² n) time. The dynamic motion planning problem is then solved in O(n²(n) log³ n) time. We also show that these results are close to optimal.