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Graph separators: a parameterized view
 Journal of Computer and System Sciences
, 2001
"... Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. ..."
Abstract

Cited by 30 (14 self)
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Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p
Shortest path queries in planar graphs in constant time
 In STOC’03
, 2003
"... We present a new algorithm for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(V ) time a data structure, which allows to check in O(1) time whether two given vertices are distant by at most k in G and if so ..."
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Cited by 6 (2 self)
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We present a new algorithm for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(V ) time a data structure, which allows to check in O(1) time whether two given vertices are distant by at most k in G and if so a shortest path between them is returned. This significantly improves the previous result of D. Eppstein [5] where after a linear preprocessing the queries are answered in O(log V ) time. Our approach can be applied to compute the girth of a planar graph and a corresponding shortest cycle in O(V ) time provided that the constant bound on the girth is known. Our results can be easily generalized to other wide classes of graphs – for instance we can take graphs embeddable in a surface of bounded genus or graphs of bounded treewidth. Categories and Subject Descriptors G.2.2 [Discrete Mathematics]: Graph Theory—graph algorithms, path and circuit problems
Finding shortest contractible and shortest separating cycles in embedded graphs
 ACM Transactions on Algorithms
"... embedded graphs ∗ ..."
Oracles for bounded length shortest paths in planar graphs
 ACM Trans. Algorithms
"... We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(V ) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so ..."
Abstract

Cited by 4 (0 self)
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We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(V ) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so a shortest path between them is returned. Graph G can be undirected as well as directed. Our data structure works in fully dynamic environment. It can be updated in O(1) time after removing an edge or a vertex while updating after an edge insertion takes polylogarithmic amortized time. Besides deleting elements one can also disable ones for some time. It is motivated by a practical situation where nodes or links of a network may be temporarily out of service. Our results can be easily generalized to other wide classes of graphs – for instance we can take any minorclosed family of graphs.
COMPUTING THE GIRTH OF A PLANAR GRAPH IN O(N log N) TIME ∗
"... Abstract. We give an O(n log n) algorithm for computing the girth (shortest cycle) of an undirected nvertex planar graph. Our solution extends to any graph of bounded genus. This improves upon the best previously known algorithms for this problem. Key words. Girth, shortest cycle, planar graph, gra ..."
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Abstract. We give an O(n log n) algorithm for computing the girth (shortest cycle) of an undirected nvertex planar graph. Our solution extends to any graph of bounded genus. This improves upon the best previously known algorithms for this problem. Key words. Girth, shortest cycle, planar graph, graphs of bounded genus AMS subject classifications. 05C38, 68R10
in embedded graphs ∗
, 2008
"... We give a polynomialtime algorithm to find a shortest contractible cycle (i.e. a closed walk without repeated vertices) in a graph embedded in a surface. This answers a question posed by Hutchinson. In contrast, we show that finding a shortest contractible cycle through a given vertex is NPhard. W ..."
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We give a polynomialtime algorithm to find a shortest contractible cycle (i.e. a closed walk without repeated vertices) in a graph embedded in a surface. This answers a question posed by Hutchinson. In contrast, we show that finding a shortest contractible cycle through a given vertex is NPhard. We also show that finding a shortest separating cycle in an embedded graph is NPhard. This answers a question posed by Mohar and Thomassen. 1