231,583 research outputs found

    An Upper Bound on the Size of Obstructions for Bounded Linear Rank-Width

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    We provide a doubly exponential upper bound in pp on the size of forbidden pivot-minors for symmetric or skew-symmetric matrices over a fixed finite field F\mathbb{F} of linear rank-width at most pp. As a corollary, we obtain a doubly exponential upper bound in pp on the size of forbidden vertex-minors for graphs of linear rank-width at most pp. This solves an open question raised by Jeong, Kwon, and Oum [Excluded vertex-minors for graphs of linear rank-width at most kk. European J. Combin., 41:242--257, 2014]. We also give a doubly exponential upper bound in pp on the size of forbidden minors for matroids representable over a fixed finite field of path-width at most pp. Our basic tool is the pseudo-minor order used by Lagergren [Upper Bounds on the Size of Obstructions and Interwines, Journal of Combinatorial Theory Series B, 73:7--40, 1998] to bound the size of forbidden graph minors for bounded path-width. To adapt this notion into linear rank-width, it is necessary to well define partial pieces of graphs and merging operations that fit to pivot-minors. Using the algebraic operations introduced by Courcelle and Kant\'e, and then extended to (skew-)symmetric matrices by Kant\'e and Rao, we define boundaried ss-labelled graphs and prove similar structure theorems for pivot-minor and linear rank-width.Comment: 28 pages, 1 figur

    On the efficiency of estimating penetrating rank on large graphs

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    P-Rank (Penetrating Rank) has been suggested as a useful measure of structural similarity that takes account of both incoming and outgoing edges in ubiquitous networks. Existing work often utilizes memoization to compute P-Rank similarity in an iterative fashion, which requires cubic time in the worst case. Besides, previous methods mainly focus on the deterministic computation of P-Rank, but lack the probabilistic framework that scales well for large graphs. In this paper, we propose two efficient algorithms for computing P-Rank on large graphs. The first observation is that a large body of objects in a real graph usually share similar neighborhood structures. By merging such objects with an explicit low-rank factorization, we devise a deterministic algorithm to compute P-Rank in quadratic time. The second observation is that by converting the iterative form of P-Rank into a matrix power series form, we can leverage the random sampling approach to probabilistically compute P-Rank in linear time with provable accuracy guarantees. The empirical results on both real and synthetic datasets show that our approaches achieve high time efficiency with controlled error and outperform the baseline algorithms by at least one order of magnitude

    The First Order Definability of Graphs with Separators via the Ehrenfeucht Game

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    We say that a first order formula Φ\Phi defines a graph GG if Φ\Phi is true on GG and false on every graph GG' non-isomorphic with GG. Let D(G)D(G) be the minimal quantifier rank of a such formula. We prove that, if GG is a tree of bounded degree or a Hamiltonian (equivalently, 2-connected) outerplanar graph, then D(G)=O(logn)D(G)=O(\log n), where nn denotes the order of GG. This bound is optimal up to a constant factor. If hh is a constant, for connected graphs with no minor KhK_h and degree O(n/logn)O(\sqrt n/\log n), we prove the bound D(G)=O(n)D(G)=O(\sqrt n). This result applies to planar graphs and, more generally, to graphs of bounded genus.Comment: 17 page

    Deciding first-order properties of nowhere dense graphs

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    Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.Comment: 30 page

    Minimum Rank of Graphs with Loops

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    A loop graph S is a finite undirected graph that allows loops but does not allow multiple edges. The set S(S) of real symmetric matrices associated with a loop graph of order n is the set of symmetric matrices A = [a(ij)] is an element of R-nxn such that a(ij) not equal 0 if and only if ij is an element of E(S). The minimum (maximum) rank of a loop graph is the minimum (maximum) of t he ranks of the matrices in S(S). Loop graphs having minimum rank at most two are characterized (by forbidden induced subgraphs and graph complements) and loop graphs having minimum rank equal to the order of the graph are characterized. A Schur complement reduction technique is used to determine the minimum ranks of cycles with various loop configurations; the minimum ranks of complete graphs and paths with various configurations of loops are also determined. Unlike simple graphs, loop graphs can have maximum rank less than the order of the graph. Some results are presented on maximum rank and which ranks between minimum and maximum can be realized. Interesting open questions remain

    Graphs and Number Theory.

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    In the 1930\u27s, L. Redei and H. Reichardt used certain matrices to aid in the determination of the structure of ideal class groups of quadratic number fields. This is a classical number theoretic problem which in general presents diffculties. Ideal class groups are finite abelian groups, and it is a result of Gauss that allows us to determine their 2-rank, in other words the number of cyclic factors of even order. Redei and Reichardt worked on determining the 4-rank, the number of factors of order divisible by 4. Later, the classical study of circulant graphs was utilized to further help this determination. In particular, if we relate a certain circulant graph G to a quadratic number field, then the number of Eulerian Vertex Decompositions of G is closely related to the 4-rank of the ideal class group of the quadratic number field. Circulant graphs however become large rather quickly. Recently, P. E. Conner and J. Hurrelbrink developed the concept of quotient graphs. These are significantly smaller graphs, yet by analyzing their structure, one can determine much of the same number theoretic information, including the 4-rank of the ideal class group of the related quadratic number field, as one can from the underlying circulant graph. Formal quotient graphs are a generalization of quotient graphs and are a useful tool in determining how many graphs on a given number of vertices can be realized as quotient graphs. In Chapter 1, we develop the background information on circulant graphs and explore their structure. We then utilize circulant graphs in Chapter 2 with the development of quotient graphs. In this chapter we determine exactly which graphs on 2, 3, 4, 5 and 7 vertices are quotient graphs. Finally in Chapter 3, we develop the concept of formal quotient graphs as a generalization of quotient graphs. By analyzing the general situation, we are able to count how many formal quotient graphs there are on 11, 13, 17 and 19 vertices and realize many of these graphs as actual quotient graphs
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