Sang‐il Oum

Rank-width was defined by Oum and Seymour [2006] to investigate clique-width. They constructed an algorithm that either outputs a rank-decomposition of width at most f ( k ) for some function f or confirms that rank-width is larger than k in time O (| V | 9 log | V |) for an input graph G = ( V , E ) and a fixed k . We develop three separate algorithms of this kind with faster running time. We construct an O (| V | 4 )-time algorithm with f ( k ) = 3 k + 1 by constructing a subroutine for the pr

Robertson and Seymour [J. Combin. Theory Ser. B, 48 (1990), pp. 227–254] proved that graphs of bounded tree-width are well-quasi-ordered by the graph minor relation. By extending their arguments, Geelen, Gerards, and Whittle [J. Combin. Theory Ser. B, 84 (2002), pp. 270–290] proved that binary matroids of bounded branch-width are well-quasi-ordered by the matroid minor relation. We prove another theorem of this kind in terms of rank-width and vertex-minors. For a graph $G=(V,E)$ and a vertex v o

Abstract We prove that the rank‐width of the incidence graph of a graph G is either equal to or exactly one less than the branch‐width of G , unless the maximum degree of G is 0 or 1. This implies that rank‐width of a graph is less than or equal to branch‐width of the graph unless the branch‐width is 0. Moreover, this inequality is tight. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 239–244, 2008

Abstract We prove that, for a fixed bipartite circle graph H , all line graphs with sufficiently large rank‐width (or clique‐width) must have a pivot‐minor isomorphic to H . To prove this, we introduce graphic delta‐matroids. Graphic delta‐matroids are minors of delta‐matroids of line graphs and they generalize graphic and cographic matroids. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 183–203, 2009

Lovász and Plummer conjectured that there exists a fixed positive constant $c$ such that every cubic $n$-vertex graph with no cutedge has at least $2^{cn}$ perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic $n$-vertex graph with no cutedge has more than $2^{n/12}$ perfect matchings, thus verifying the conjecture for claw-free graphs.

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph G=(V,E) and a subset A of E we let λG(A) be the number of vertices incident with an edge in A and an edge in E∖A. For a subset X of V, let ρG(X) be the rank of the adjacency matrix between X and V∖X over the binary field. We prove that