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Localized and compact datastructure for comparability graphs
, 2009
"... We show that every comparability graph of any twodimensional poset over n elements (a.k.a. permutation graph) can be preprocessed in O(n) time, if two linear extensions of the poset are given, to produce an O(n) space datastructure supporting distance queries in constant time. The datastructure i ..."
Abstract

Cited by 12 (5 self)
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We show that every comparability graph of any twodimensional poset over n elements (a.k.a. permutation graph) can be preprocessed in O(n) time, if two linear extensions of the poset are given, to produce an O(n) space datastructure supporting distance queries in constant time. The datastructure is localized and given as a distance labeling, that is each vertex receives a label of O(log n) bits so that distance queries between any two vertices are answered by inspecting their labels only. This result improves the previous scheme due to Katz, Katz and Peleg [M. Katz, N.A. Katz, D. Peleg, Distance labeling schemes for wellseparated graph classes, Discrete Applied Mathematics 145 (2005) 384–402] by a log n factor. As a byproduct, our datastructure supports allpair shortestpath queries in O(d) time for distanced pairs, and so identifies in constant time the first edge along a shortest path between any source and destination. More fundamentally, we show that this optimal space and time datastructure cannot be extended for higher dimension posets. More precisely, we prove that for comparability graphs of threedimensional posets, every distance labeling scheme requires Ω(n 1/3) bit labels.
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 ..."
Abstract

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.
Coloring trianglefree graphs on surfaces (Extended Abstract)
"... Gimbel and Thomassen asked whether 3colorability of a trianglefree graph drawn on a fixed surface can be tested in polynomial time. We settle the question by giving a lineartime algorithm for every surface which combined with previous results gives a lineartime algorithm to compute the chromatic ..."
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Gimbel and Thomassen asked whether 3colorability of a trianglefree graph drawn on a fixed surface can be tested in polynomial time. We settle the question by giving a lineartime algorithm for every surface which combined with previous results gives a lineartime algorithm to compute the chromatic number of such graphs. Our algorithm is based on a structure theorem that for a trianglefree graph drawn on a surface Σ guarantees the existence of a subgraph H, whose size depends only on Σ, such that there is an easy test whether a 3coloring of H extends to a 3coloring of G. The test is based on a topological obstruction, called the “winding number” of a 3coloring. To prove the structure theorem we make use of disjoint paths with specified ends to find a 3coloring. If the input trianglefree graph G drawn in Σ is 3colorable we can find a 3coloring in quadratic time, and if G quadrangulates Σ then we can find the 3coloring in linear time. The latter algorithm requires two ingredients that may be of independent interest: a generalization of a data structure of Kowalik and Kurowski to weighted graphs and a speedup of a disjoint paths algorithm of Robertson and Seymour to linear time.