424 research outputs found

    Rerouting shortest paths in planar graphs

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    A rerouting sequence is a sequence of shortest st-paths such that consecutive paths differ in one vertex. We study the the Shortest Path Rerouting Problem, which asks, given two shortest st-paths P and Q in a graph G, whether a rerouting sequence exists from P to Q. This problem is PSPACE-hard in general, but we show that it can be solved in polynomial time if G is planar. To this end, we introduce a dynamic programming method for reconfiguration problems.Comment: submitte

    The List Coloring Reconfiguration Problem for Bounded Pathwidth Graphs

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    We study the problem of transforming one list (vertex) coloring of a graph into another list coloring by changing only one vertex color assignment at a time, while at all times maintaining a list coloring, given a list of allowed colors for each vertex. This problem is known to be PSPACE-complete for bipartite planar graphs. In this paper, we first show that the problem remains PSPACE-complete even for bipartite series-parallel graphs, which form a proper subclass of bipartite planar graphs. We note that our reduction indeed shows the PSPACE-completeness for graphs with pathwidth two, and it can be extended for threshold graphs. In contrast, we give a polynomial-time algorithm to solve the problem for graphs with pathwidth one. Thus, this paper gives precise analyses of the problem with respect to pathwidth

    Rerouting Planar Curves and Disjoint Paths

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    In this paper, we consider a transformation of k disjoint paths in a graph. For a graph and a pair of k disjoint paths ? and ? connecting the same set of terminal pairs, we aim to determine whether ? can be transformed to ? by repeatedly replacing one path with another path so that the intermediates are also k disjoint paths. The problem is called Disjoint Paths Reconfiguration. We first show that Disjoint Paths Reconfiguration is PSPACE-complete even when k = 2. On the other hand, we prove that, when the graph is embedded on a plane and all paths in ? and ? connect the boundaries of two faces, Disjoint Paths Reconfiguration can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint s-t paths as a variant. We show that the disjoint s-t paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is PSPACE-complete in general

    Colored Non-Crossing Euclidean Steiner Forest

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    Given a set of kk-colored points in the plane, we consider the problem of finding kk trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For k=1k=1, this is the well-known Euclidean Steiner tree problem. For general kk, a kρk\rho-approximation algorithm is known, where ρ1.21\rho \le 1.21 is the Steiner ratio. We present a PTAS for k=2k=2, a (5/3+ε)(5/3+\varepsilon)-approximation algorithm for k=3k=3, and two approximation algorithms for general~kk, with ratios O(nlogk)O(\sqrt n \log k) and k+εk+\varepsilon

    Recognizing and Drawing IC-planar Graphs

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    IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph GG with nn vertices, we present an O(n)O(n)-time algorithm that computes a straight-line drawing of GG in quadratic area, and an O(n3)O(n^3)-time algorithm that computes a straight-line drawing of GG with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar

    Reconfiguration on sparse graphs

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    A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions S and T of size k, whether it is possible to transform S into T by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that ISR is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that ISR is fixed-parameter tractable on planar graphs and graphs of bounded degree. For DSR, we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowhere-dense graphs

    Upper and Lower Bounds on Long Dual-Paths in Line Arrangements

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    Given a line arrangement A\cal A with nn lines, we show that there exists a path of length n2/3O(n)n^2/3 - O(n) in the dual graph of A\cal A formed by its faces. This bound is tight up to lower order terms. For the bicolored version, we describe an example of a line arrangement with 3k3k blue and 2k2k red lines with no alternating path longer than 14k14k. Further, we show that any line arrangement with nn lines has a coloring such that it has an alternating path of length Ω(n2/logn)\Omega (n^2/ \log n). Our results also hold for pseudoline arrangements.Comment: 19 page
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