O-joung Kwon

Linear Rank-Width of Distance-Hereditary Graphs I. A Polynomial-Time Algorithm

Linear rank-width is a linearized variation of rank-width, and it is deeply related to matroid path-width. In this paper, we show that the linear rank-width of every n-vertex distance-hereditary graph, equivalently a graph of rank-width at most 1, can be computed in time

Neighborhood Complexity and Kernelization for Nowhere Dense Classes of Graphs

{\mathcal {O}}(n^2\cdot \log _2 n)

Characterizing width two for variants of treewidth

O(n2·log2n), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. As a corollary, we show that the path-width of every n-element matroid of branch-width at most 2 can be computed in time

Linear Rank-Width of Distance-Hereditary Graphs

{\mathcal {O}}(n^2\cdot \log _2 n)

Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions

O(n2·log2n), provided that the matroid is given by its binary representation. To establish this result, we present a characterization of the linear rank-width of distance-hereditary graphs in terms of their canonical split decompositions. This characterization is similar to the known characterization of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex separation and search number of a graph. Inf. Comput., 113(1):50–79, 1994]. However, different from forests, it is non-trivial to relate substructures of the canonical split decomposition of a graph with some substructures of the given graph. We introduce a notion of ‘limbs’ of canonical split decompositions, which correspond to certain vertex-minors of the original graph, for the right characterization.

Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors

In 2013 Belmonte and Vatshelle used mim-width, a graph parameter bounded on interval graphs and permutation graphs that strictly generalizes clique-width, to explain existing algorithms for many domination-type problems, known as LC-VSVP problems. We focus on chordal graphs and co-comparability graphs, that strictly contain interval graphs and permutation graphs respectively. First, we show that mim-width is unbounded on these classes, thereby settling an open problem from 2012. Then, we introduce two graphs \(K_t \boxminus K_t\) and \(K_t \boxminus S_t\) to restrict these graph classes, obtained from the disjoint union of two cliques of size t, and one clique of size t and one independent set of size t respectively, by adding a perfect matching. We prove that \((K_t \boxminus S_t)\)-free chordal graphs have mim-width at most \(t-1\), and \((K_t \boxminus K_t)\)-free co-comparability graphs have mim-width at most \(t-1\). From this, we obtain several algorithmic consequences, for instance, while Dominating Set is NP-complete on chordal graphs, it can be solved in time \(\mathcal {O}(n^{t})\) on chordal graphs where t is the maximum among induced subgraphs \(K_t \boxminus S_t\) in the given graph. We also show that classes restricted in this way have unbounded rank-width which validates our approach.