We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2, ... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in M_j, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour's theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum's theorem for graphs of bounded rank-width with respect to pivot-minors.
We study the commutation relations and normal ordering between families of operators on symmetric functions. These operators can be naturally defined by the operations of multiplication, Kronecker product, and their adjoints. As applications we give a new proof of the skew Littlewood-Richardson rule and prove an identity about the Kronecker product with a skew Schur function.
Inspired by the works of Dewar, Murty and Kotěšovec, we establish some useful theorems for asymptotic formulas. As an application, we obtain asymptotic formulas for the numbers of skew plane partitions and cylindric partitions. We prove that the order of the asymptotic formula for the skew plane partitions of fixed width depends only on the width of the region, not on the profile (the skew zone) itself, while this is not true for cylindric partitions.
Lattice path matroids form a subclass of transversal matroids and were introduced by Bonin, de Mier and Noy. Transversal matroids are not well-quasi-ordered, even when the branch-width is restricted. Though lattice path matroids are not well-quasi-ordered, we prove that lattice path matroids of bounded branch-width are well-quasi-ordered.
We present a hereditary class of graphs of unbounded clique-width which is well-quasi-ordered by the induced subgraph relation. This result provides a negative answer to the question asked by Daligault, Rao and Thomassé in ("Well-quasi-order of relabel functions", Order, 27(3):301--315, 2010).
In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field $\mathbf{F}$, any infinite sequence $M_1,M_2,...$ of (skew) symmetric matrices over $\mathbf{F}$ of bounded $\mathbf{F}$-rank-width has a pair $i< j$, such that $M_i$ is isomorphic to a principal submatrix of a principal pivot transform of $M_j$. We generalise this result to $σ$-symmetric matrices introduced by Rao and myself in [The Rank-Width of Edge-Coloured Graphs, arXiv:0709.1433v4]. (Skew) symmetric matrices are special cases of $σ$-symmetric matric…
There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. We consider the posets that result from ordering skew diagrams according to Schur positivity, before focussing on the convex subposets corresponding to ribbons. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those whose expansion as a linear combination of Schur functions has all coefficients either zero or one. In particular, we show that the poset that…
Daligault, Rao and Thomassé asked whether a hereditary class of graphs well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this is not true for classes defined by infinitely many forbidden induced subgraphs. However, in the case of finitely many forbidden induced subgraphs the question remains open and we conjecture that in this case the answer is positive. The conjecture is known to hold for classes of graphs defined by a single forbidden induced subgraph $H$, as such graphs are well-quasi-ordered and are of bounded cli…
We provide a doubly exponential upper bound in $p$ on the size of forbidden pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field $\mathbb{F}$ of linear rank-width at most $p$. As a corollary, we obtain a doubly exponential upper bound in $p$ on the size of forbidden vertex-minors for graphs of linear rank-width at most $p$. This solves an open question raised by Jeong, Kwon, and Oum [Excluded vertex-minors for graphs of linear rank-width at most $k$. European J. Combin., 41:242--257, 2014]. We also give a doubly exponential upper bound in $p$ on the size of forbidden…
We introduce the notion of z-topological orderings for digraphs. We prove that given a digraph G on n vertices admitting a z-topological order- ing, together with such an ordering, one may count the number of subgraphs of G that at the same time satisfy a monadic second order formula ϕ and are the union of k directed paths, in time f (ϕ, k, z) * n^O(k*z) . Our result implies the polynomial time solvability of many natural counting problems on digraphs admitting z-topological orderings for constant values of z and k. Concerning the relationship between z-topological orderability and other digra…
Daligault, Rao and Thomassé asked whether every hereditary graph class that is well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev (JCTB 2017+) gave a negative answer to this question, but their counterexample is a class that can only be characterised by infinitely many forbidden induced subgraphs. This raises the issue of whether the question has a positive answer for finitely defined hereditary graph classes. Apart from two stubborn cases, this has been confirmed when at most two induced subgraphs $H_1,H_2$ are forbidden. We confirm it for…
The class of bipartite permutation graphs enjoys many nice and important properties. In particular, this class is critically important in the study of clique- and rank-width of graphs, because it is one of the minimal hereditary classes of graphs of unbounded clique- and rank-width. It also contains a number of important subclasses, which are critical with respect to other parameters, such as graph lettericity or shrub-depth, and with respect to other notions, such as well-quasi-ordering or complexity of algorithmic problems. In the present paper we identify critical subclasses of bipartite pe…
This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs. Several open problems about hereditary classes of relational structures which have been considered over the years have positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and M…
We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing …
We recently introduced the graph invariant twin-width, and showed that first-order model checking can be solved in time $f(d,k)n$ for $n$-vertex graphs given with a witness that the twin-width is at most $d$, called $d$-contraction sequence or $d$-sequence, and formulas of size $k$ [Bonnet et al., FOCS '20]. The inevitable price to pay for such a general result is that $f$ is a tower of exponentials of height roughly $k$. In this paper, we show that algorithms based on twin-width need not be impractical. We present $2^{O(k)}n$-time algorithms for $k$-Independent Set, $r$-Scattered Set, $k$-Cli…
