String graphs and incomparability graphs

Comparability graphs and cover-incomparability graphs (C-I graphs) are two interesting classes of graphs from posets. Comparability graph of a poset $P=(V,\le )$ is a graph with vertex set V and two vertices u and v are adjacent in V if u and v are ...

An r-quasiplanar graph is a graph drawn in the plane with no r pairwise crossing edges. Let $s\ge 3$ be an integer and $r=2^s$. We prove that there is a constant C such that every r-quasiplanar graph with $n\ge r$ vertices has at most $n{\left(C{s}^{-1}logn\right)}^{2s-4}$ edges.

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A clique-transversal of a graph G is a subset of vertices that meets all the cliques of G. A clique-independent set is a collection of pairwise vertex-disjoint cliques. The clique-transversal number and clique-independence number of G are the sizes of a ...