A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. IC-planarity specializes both NIC-planarity, which allows a pair of crossing edges to share at most one vertex, and 1-planarity, where each edge may be crossed at most once. We show that there are infinitely maximal IC-planar graphs wi...

We present an empirical analysis of some heuristics for the graph thickness problem, i.e., decomposing a graph into the minimum number of planar subgraphs. The problem has applications in database systems, e.g., the layout of E-R diagrams. The heuristics are based on some algorithms for nding a maximal planar subgraph of a nonplanar graph. Empirical results show that a randomized heuristic is a...

A clique-coloring of a given graph G is a coloring of the vertices of G such that no maximal clique of size at least two is monocolored. The clique-chromatic number of G is the least number of colors for which G admits a clique-coloring. It has been proved that every planar graph is 3-clique colorable and every claw-free planar graph, different from an odd cycle, is 2-clique colorable. In this ...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

We consider graph drawing algorithms for learning spaces, a type of st-oriented partial cube derived from antimatroids and used to model states of knowledge of students. We show how to draw any st-planar learning space so all internal faces are convex quadrilaterals with the bottom side horizontal and the left side vertical, with one minimal and one maximal vertex. Conversely, every such drawin...

We present evidence that loop amplitudes in maximally supersymmetric (N=4) Yang-Mills theory (SYM) beyond the planar limit share some of the remarkable structures of the planar theory. In particular, we show that through two loops, the four-particle amplitude in full N=4 SYM has only logarithmic singularities and is free of any poles at infinity--properties closely related to uniform transcende...

We say that a convex planar billiard table $$B$$ is $$C^{2}$$ -stably expansive on fixed open subset $$U$$ of the phase space if its map $$f_{B}$$ maximal invariant set $$\Lambda_{B,U}=\bigcap_{n\in\mathbb{Z}}f^{n}_{B}(U)$$ , and this property holds under -perturbations table. In note we prove for such billiards closure periodic points in $$\Lambda_{B,U}$$ uniformly hyperbolic. addition, show a...

The problem of finding the maximum-weight, planar subgraph of a finite, simple graph with nonnegative real edge weights is well known in industrial and electrical engineering, systems biology, sociology and finance. As the problem is known to be NP-hard, much research effort has been devoted over the years to attempt to improve a given approximate solution to the problem by using local moves ap...

In this paper, we study the k-tree partition problem which is a partition of the set of edges of a graph into k edge-disjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) area planar straight line drawing of maximal planar graphs using Schnyder’s realizers [15], which a...

In 1985 we isolated a new vascular anticoagulant protein VAC alpha, now called annexin V, with a high binding affinity (Kd less than 10(-10) M) for phospholipids. Its anticoagulant effect was attributed to displacement of coagulation factors from the phospholipid membrane. The present study demonstrates that the inhibition of prothrombinase activity by annexin V strongly depends on the curvatur...