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Many distances in planar graphs
 In SODA ’06: Proc. 17th Symp. Discrete algorithms
, 2006
"... We show how to compute in O(n 4/3 log 1/3 n+n 2/3 k 2/3 logn) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/logn) 5/6 ≤ k ≤ n 2 /log 6 n. As an application, we speed up previous algorithms for computing the dilatio ..."
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Cited by 18 (5 self)
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We show how to compute in O(n 4/3 log 1/3 n+n 2/3 k 2/3 logn) time the distance between k given pairs of vertices of a planar graph G with n vertices. This improves previous results whenever (n/logn) 5/6 ≤ k ≤ n 2 /log 6 n. As an application, we speed up previous algorithms for computing the dilation of geometric planar graphs. 1
Deciding firstorder properties for sparse graphs
"... We present a lineartime algorithm for deciding firstorder logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded treewidth, all proper minorclosed classes of graphs, graphs of bounded degree, graphs with no sub ..."
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Cited by 12 (1 self)
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We present a lineartime algorithm for deciding firstorder logic (FOL) properties in classes of graphs with bounded expansion. Many natural classes of graphs have bounded expansion: graphs of bounded treewidth, all proper minorclosed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We also develop an almost lineartime algorithm for deciding FOL properties in classes of graphs with locally bounded expansion; those include classes of graphs with locally bounded treewidth or locally excluding a minor. More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a lineartime initialization the data structure allows us to test an FOL property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their addition results in a graph in the class. In addition, we design a dynamic data structure for testing existential properties or the existence of short paths between prescribed vertices in such classes of graphs. All our results also hold for relational structures and are based on the seminal result of Nesetril and Ossona de Mendez on the existence of low treedepth colorings.
Fast 3Coloring TriangleFree Planar Graphs
 12th Annual European Symposium (ESA’04), Lecture Notes In Computer Science, (c
, 2004
"... Abstract. We show the first o(n 2) algorithm for coloring vertices of trianglefree planar graphs using three colors. The time complexity of the algorithm is O(nlog n). Our approach can be also used to designO(npolylog n)time algorithms for two other similar coloring problems. A remarkable ingredien ..."
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Cited by 7 (0 self)
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Abstract. We show the first o(n 2) algorithm for coloring vertices of trianglefree planar graphs using three colors. The time complexity of the algorithm is O(nlog n). Our approach can be also used to designO(npolylog n)time algorithms for two other similar coloring problems. A remarkable ingredient of our algorithm is the data structure processing short path queries introduced recently in [9]. In this paper we show how to adapt it to the fully dynamic environment where edge insertions and deletions are allowed. 1
Testing Planarity of Partially Embedded Graphs
, 2009
"... We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in man ..."
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Cited by 5 (2 self)
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We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomialtime solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the “oncas” behaviour – obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach lineartime complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NPhard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.
Approximation Scheme for Lowest Outdegree Orientation and Graph Density Measures
"... Abstract. We deal with the problem of finding such an orientation of a given graph that the largest number of edges leaving a vertex (called the outdegree of the orientation) is small. For any ε ∈ (0, 1) we show an Õ(E(G)/ε) time algorithm3 which finds an orientation of an input graph G with outde ..."
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Cited by 2 (0 self)
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Abstract. We deal with the problem of finding such an orientation of a given graph that the largest number of edges leaving a vertex (called the outdegree of the orientation) is small. For any ε ∈ (0, 1) we show an Õ(E(G)/ε) time algorithm3 which finds an orientation of an input graph G with outdegree at most ⌈(1 + ε)d ∗ ⌉, where d ∗ is the maximum density of a subgraph of G. It is known that the optimal value of orientation outdegree is ⌈d ∗ ⌉. Our algorithm has applications in constructing labeling schemes, introduced by Kannan et al. in [18] and in approximating such graph density measures as arboricity, pseudoarboricity and maximum density. Our results improve over the previous, 2approximation algorithms by Aichholzer et al. [1] (for orientation / pseudoarboricity), by Arikati et al. [3] (for arboricity) and by Charikar [5] (for maximum density). 1