Kernelization is a powerful tool to obtain fixed-parameter tractable algorithms. Recent breakthroughs show that many graph problems admit small polynomial kernels when restricted to sparse graph classes such as planar graphs, bounded-genus graphs or H-minor-free graphs. We consider the intersection graphs of (unit) disks in the plane, which can be arbitrarily dense but do exhibit some geometric...

In this paper we extend the work of Rautenbach and Szwarcfiter [8] by giving a structural characterization of graphs that can be represented by the intersection of unit intervals that may or may not contain their endpoints. A characterization was proved independently by Joos in [6], however our approach provides an algorithm that produces such a representation, as well as a forbidden graph char...

A rectangular drawing of a planar graph G is in which vertices are mapped to grid points, edges horizontal and vertical straight-line segments, faces drawn as rectangles. Sometimes this latter constraint relaxed for the outer face. In paper, we study drawings have unit length. We show complexity dichotomy problem deciding existence unit-length drawing, depending on whether face must also be rec...

A graph is probe (unit) interval if its vertices can be partitioned into two sets: a set of probe vertices and a set of nonprobe vertices, so that the set of nonprobe vertices is a stable set and it is possible to obtain a (unit) interval graph by adding edges with both endpoints in the set of nonprobe vertices. Probe (unit) interval graphs form a superclass of (unit) interval graphs. Probe int...