Crossing-number critical graphs have bounded path-width

Crossing-number critical graphs have bounded path-width

The crossing number of a graph G, denoted by cr(G), is defined as the smallest possible number of edge-crossings in a drawing of G in the plane. A graph G is crossing-critical if cr(G − e) < cr(G) for all edges e of G. We prove that crossing-critical graphs have “bounded path-width” (by a function of the crossing number), which roughly means that such graphs are made up of small pieces joined i...

Crossing-Critical Graphs and Path-Width

Crossing Number for Graphs with Bounded~Pathwidth

The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions,...

Gem- And Co-Gem-Free Graphs Have Bounded Clique-Width

The P4 is the induced path of four vertices. The gem consists of a P4 with an additional universal vertex being completely adjacent to the P4, and the co-gem is its complement graph. Gemand co-gem-free graphs generalize the popular class of cographs (i. e. P4-free graphs). The tree structure and algebraic generation of cographs has been crucial for several concepts of graph decomposition such a...

On the Queue-Number of Graphs with Bounded Tree-Width

A queue layout of a graph consists of a linear order of the vertices and an assignment of the edges to queues, such that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called its queue-number. We show that for each k > 0, graphs with tree-width at most k have queuenumber at most 2k − 1. This improves upon double exponential upper b...