We introduce the family of k-gap-planar graphs for k ≥ 0, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most k of its crossings. This definition is motivated by applications in edge casing, as a k-gap-planar graph can be drawn crossing-free after introducing at most k local gaps per edge. We present results on t...

In 1979, Nishizeki and Baybars showed that every planar graph with minimum degree 3 has a matching of size n + c (where the constant depends on connectivity), even better bounds hold for graphs 4 5. this paper, we investigate similar matching-bounds 1-planar graphs, is, can be drawn such edge at most one crossing. We show least 1 7 12 , is tight some graphs. provide 5, while case 6 remains open.

We introduce Markov Random Geometric Graphs (MRGGs), a growth model for temporal dynamic networks. It is based on Markovian latent space dynamic: consecutive points are sampled the Euclidean Sphere using an unknown kernel; and two nodes connected with probability depending function of their geodesic distance. More precisely, at each stamp-time k we add point Xk by jumping from previous one Xk−1...

Graphs are an important model for the representation of structural information between objects. One identifies objects and nodes as well as a binary relation between objects and edges. Graphs have many uses, e. g., in social sciences, life sciences and engineering. There are two primary representations: abstract and visual. The abstract representation is well suited for processing graphs by com...

We study the maximum flow problem in directed H-minorfree graphs where H can be drawn in the plane with one crossing. If a structural decomposition of the graph as a clique-sum of planar graphs and graphs of constant complexity is given, we show that a maximum flow can be computed in O(n logn) time. In particular, maximum flows in directed K3,3-minor-free graphs and directed K5-minor-free graph...

Chordal graphs are intersection graphs of subtrees of a tree T . We investigate the complexity of the partial representation extension problem for chordal graphs. A partial representation specifies a tree T ′ and some pre-drawn subtrees of T . It asks whether it is possible to construct a representation inside a modified tree T which extends the partial representation (i.e, keeps the pre-drawn ...