42 research outputs found

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    EPG-representations with small grid-size

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    In an EPG-representation of a graph GG each vertex is represented by a path in the rectangular grid, and (v,w)(v,w) is an edge in GG if and only if the paths representing vv an ww share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x- and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters. We show that there are mm-edge graphs that require a grid of area Ω(m)\Omega(m) in any variant of EPG-representations. Similarly there are pathwidth-kk graphs that require height Ω(k)\Omega(k) and area Ω(kn)\Omega(kn) in any variant of EPG-representations. We prove a matching upper bound of O(kn)O(kn) area for all pathwidth-kk graphs in the strongest model, the one where edges are required to be both x- and y-monotone. Thus in this strongest model, the result implies, for example, O(n)O(n), O(nlogn)O(n \log n) and O(n3/2)O(n^{3/2}) area bounds for bounded pathwidth graphs, bounded treewidth graphs and all classes of graphs that exclude a fixed minor, respectively. For the model with no restrictions on the monotonicity of the edges, stronger results can be achieved for some graph classes, for example an O(n)O(n) area bound for bounded treewidth graphs and O(nlog2n)O(n \log^2 n) bound for graphs of bounded genus.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

    A tourist guide through treewidth

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    A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find tree-decompositions, algorithms that use tree-decompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography

    Approximating pathwidth for graphs of small treewidth

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    We describe a polynomial-time algorithm which, given a graph GG with treewidth tt, approximates the pathwidth of GG to within a ratio of O(tlogt)O(t\sqrt{\log t}). This is the first algorithm to achieve an f(t)f(t)-approximation for some function ff. Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least th+2th+2 has treewidth at least tt or contains a subdivision of a complete binary tree of height h+1h+1. The bound th+2th+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (with c=2c=2), the following conjecture of Kawarabayashi and Rossman (SODA'18): there exists a universal constant cc such that every graph with pathwidth Ω(kc)\Omega(k^c) has treewidth at least kk or contains a subdivision of a complete binary tree of height kk. Our main technical algorithm takes a graph GG and some (not necessarily optimal) tree decomposition of GG of width tt' in the input, and it computes in polynomial time an integer hh, a certificate that GG has pathwidth at least hh, and a path decomposition of GG of width at most (t+1)h+1(t'+1)h+1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height hh. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth
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