Rank-width of random graphs

Rank-width of random graphs

Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006). We investigate the asymptotic behavior of rank-width of a random graph G(n, p). We show that, asymptotically almost surely, (i) if p ∈ (0, 1) is a constant, then rw(G(n, p)) = dn3 e −O(1), (ii) if 1 n p ≤ 1 2 , then rw(G(n, p)) = d n 3 e − o(n), (iii) if p = c/n and c > 1, then rw(G(n...

Graphs of Bounded Rank-width

We define rank-width of graphs to investigate clique-width. Rank-width is a complexity measure of decomposing a graph in a kind of tree-structure, called a rankdecomposition. We show that graphs have bounded rank-width if and only if they have bounded clique-width. It is unknown how to recognize graphs of clique-width at most k for fixed k > 3 in polynomial time. However, we find an algorithm r...

Rank-width and tree-width of H-minor-free graphs

We prove that for any fixed r ≥ 2, the tree-width of graphs not containing Kr as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for other graph classes such as Kr-minor free graphs and graphs of bounded genus.

Linear Rank-Width of Distance-Hereditary Graphs

We present a characterization of the linear rank-width of distancehereditary graphs. Using the characterization, we show that the linear rankwidth of every n-vertex distance-hereditary graph can be computed in time O(n · log(n)), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. For our characterization, we combine modifications of canonical spl...

The rank-width of Directed Graphs