We give a new proof of the fact that every planar graph is 5-choosable, and use it to show that every graph drawn in the plane so that the distance between every pair of crossings is at least 15 is 5-choosable. At the same time we may allow some vertices to have lists of size four only, as long as they are far apart and far from the crossings.
COBISS.SI-ID: 18214489
A class of graphs that lies strictly between the classes of graphs of genus (at most) $k-1$ and $k$ is studied. For a fixed orientable surface $\mathbb{S}_k$ of genus $k$, let $\mathcal{A}_{xy}^k$ be the minor-closed class of graphs with terminals $x$ and $y$ that either embed into $\mathbb{S}_{k-1}$ or admit an embedding $\Pi$ into $\mathbb{S}_k$ such that there is a $\Pi$-face where $x$ and $y$ appear twice in the alternating order. In this paper, the obstructions for the classes $\mathcal{A}_{xy}^k$ are studied. In particular, the complete list of obstructions for $\mathcal{A}_{xy}^1$ is presented.
COBISS.SI-ID: 17761369
We consider the problem of finding a 1-planar drawing for a general graph, where a 1-planar drawing is a drawing in which each edge participates in at most one crossing. Since this problem is known to be NP-hard we investigate the parameterized complexity of the problem with respect to the vertex cover number, tree-depth, and cyclomatic number. For these parameters we construct fixed-parameter tractable algorithms. However, the problem remains NP-complete for graphs of bounded bandwidth, pathwidth, or treewidth.
COBISS.SI-ID: 18208345