30,678 research outputs found

    Well-Quasi-Ordering of Matrices under Schur Complement and Applications to Directed Graphs

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    In [Rank-Width and Well-Quasi-Ordering of Skew-Symmetric or Symmetric Matrices, arXiv:1007.3807v1] Oum proved that, for a fixed finite field F\mathbf{F}, any infinite sequence M1,M2,...M_1,M_2,... of (skew) symmetric matrices over F\mathbf{F} of bounded F\mathbf{F}-rank-width has a pair i<ji< j, such that MiM_i is isomorphic to a principal submatrix of a principal pivot transform of MjM_j. We generalise this result to σ\sigma-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\sigma-symmetric matrices. As a by-product, we obtain that for every infinite sequence G1,G2,...G_1,G_2,... of directed graphs of bounded rank-width there exist a pair i<ji<j such that GiG_i is a pivot-minor of GjG_j. Another consequence is that non-singular principal submatrices of a σ\sigma-symmetric matrix form a delta-matroid. We extend in this way the notion of representability of delta-matroids by Bouchet.Comment: 35 pages. Revised version with a section for directed graph

    Are there any good digraph width measures?

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    Several different measures for digraph width have appeared in the last few years. However, none of them shares all the "nice" properties of treewidth: First, being \emph{algorithmically useful} i.e. admitting polynomial-time algorithms for all \MS1-definable problems on digraphs of bounded width. And, second, having nice \emph{structural properties} i.e. being monotone under taking subdigraphs and some form of arc contractions. As for the former, (undirected) \MS1 seems to be the least common denominator of all reasonably expressive logical languages on digraphs that can speak about the edge/arc relation on the vertex set.The latter property is a necessary condition for a width measure to be characterizable by some version of the cops-and-robber game characterizing the ordinary treewidth. Our main result is that \emph{any reasonable} algorithmically useful and structurally nice digraph measure cannot be substantially different from the treewidth of the underlying undirected graph. Moreover, we introduce \emph{directed topological minors} and argue that they are the weakest useful notion of minors for digraphs

    Digraph Complexity Measures and Applications in Formal Language Theory

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    We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former algorithm yields an O((log n)^(3/2))- approximation in polynomial time, whereas the latter yields the optimum solution, and runs in time and space O*(1.9129^n) on digraphs of maximum outdegree at most two. Regarding the star height problem, we identify a subclass of the regular languages for which we can precisely determine the computational complexity of the star height problem. Namely, the star height problem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Then we translate the algorithmic results concerning cycle rank to the bideterministic star height problem, thus giving a polynomial-time approximation as well as a reasonably fast exact exponential algorithm for bideterministic star height.Comment: 19 pages, 1 figur

    Graphs of Small Rank-width are Pivot-minors of Graphs of Small Tree-width

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    We prove that every graph of rank-width kk is a pivot-minor of a graph of tree-width at most 2k2k. We also prove that graphs of rank-width at most 1, equivalently distance-hereditary graphs, are exactly vertex-minors of trees, and graphs of linear rank-width at most 1 are precisely vertex-minors of paths. In addition, we show that bipartite graphs of rank-width at most 1 are exactly pivot-minors of trees and bipartite graphs of linear rank-width at most 1 are precisely pivot-minors of paths.Comment: 16 pages, 7 figure

    An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion

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    Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514--528, 2006]. Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an nn-vertex graph GG and a positive integer kk, we want to decide whether there is a set of at most kk vertices whose removal turns GG into a graph of linear rankwidth at most 11 and find such a vertex set if one exists. While the meta-theorem of Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved in time f(k)â‹…n3f(k)\cdot n^3 for some function ff, it is not clear whether this problem allows a running time with a modest exponential function. We first establish that LRW1-Vertex Deletion can be solved in time 8kâ‹…nO(1)8^k\cdot n^{\mathcal{O}(1)}. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define necklace graphs and investigate their structural properties. Later, we reduce the polynomial factor by refining the trivial branching step based on a cliquewidth expression of a graph, and obtain an algorithm that runs in time 2O(k)â‹…n42^{\mathcal{O}(k)}\cdot n^4. We also prove that the running time cannot be improved to 2o(k)â‹…nO(1)2^{o(k)}\cdot n^{\mathcal{O}(1)} under the Exponential Time Hypothesis assumption. Lastly, we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.Comment: 29 pages, 9 figures, An extended abstract appeared in IPEC201
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