This paper presents an eecient fully dynamic graph algorithm for maintaining the transitive closure of a directed graph. The algorithm updates the adjacency matrix of the transitive closure with each update to the graph. Hence, each reachability query of the form \Is there a directed path from i to j?" can be answered in O(1) time. The algorithm is randomized; it is correct when answering yes, ...

A general method is described which gives rise to highly symmetric tessellations of the Cantor sphere, i.e., the 2-sphere with the Cantor set removed and endowed with the hyperbolic geometry with constant negative curvature. These tessellations correspond to almost vertextransitive planar graphs with infinitely many ends. Their isometry groups have infinitely many ends and are free products wit...

Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph G = (V,E), a geodesic triangle △(x, y, z) with x, y, z ∈ V is the union P (x, y)∪P (x, z)∪P (y, z) of three shortest paths connecting these vertices. A geodesic triangle △(x, y, z) is called δ-slim if for any vertex u ∈ V on any side P (x, y) the distance from u to P (x, z)∪P (y, z) is at most δ, i.e. ...

A tournament is an oriented complete graph, and one containing no directed cycles is called transitive. A tournament T= (V,A) is called m-partition transitive if there is a partition V=X1∪· X2∪· · · ·∪· Xm such that the subtournaments induced by each Xi are all transitive, and T Contract grant sponsor: University of Dayton Research Council (to A. H. B.); Contract grant sponsor: National Science...

In this paper, present a graph cuts based geodesic active contours (GAC) approach to object segmentation problems. Our method is a combination of geodesic active contours and the optimization tool of graph cuts and differs fundamentally from traditional active contours in that it uses graph cuts to iteratively deform the contour. Consequently, it has the following advantages. 1. It has the abil...

Generalized associahedra were introduced by S. Fomin and A. Zelevinsky in connection to finite type cluster algebras. Following recent work of L. Pournin in types A and B, this paper focuses on geodesic properties of generalized associahedra. We prove that the graph diameter of the n-dimensional associahedron of type D is precisely 2n − 2 for all n greater than 1. Furthermore, we show that all ...

For a Riemannian submersion from a simple compact Lie group with a bi-invariant metric, we prove the action of its holonomy group on the fibers is transitive. As a step towards classifying Riemannian submersions with totally geodesic fibers, we consider the parameterized surface induced by lifting a base geodesic to points along a geodesic in a fiber. Such a surface is “doubly ruled” (it is rul...

Two questions are considered, namely (i) How many colors are needed for a coloring of the n-cube without monochromatic quadrangles or hexagons? We show that four colors suffice and thereby settle a problem of Erdos. (ii) Which vertex-transitive induced subgraphs does a hypercube have? An interesting graph has come up in this context: If we delete a Hamming code from the 7-cube, the resulting gr...

Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg ([14]) is whether there exists a bi-infinite geodesic in first passage percolation on the euclidean lattice of dimension at least 2. Although the answer is generally conjectured to be negative, we give a positive answer for graphs sa...

We show that an action of SL(2, p), p ≥ 7 an odd prime such that 4 6 | (p − 1), has exactly two orbital digraphs Γ1, Γ2, such that Aut(Γi) admits a complete block system B of p + 1 blocks of size 2, i = 1, 2, with the following properties: the action of Aut(Γi) on the blocks of B is nonsolvable, doubly-transitive, but not a symmetric group, and the subgroup of Aut(Γi) that fixes each block of B...