 School of Mathematics and Statistics  Research Publications
School of Mathematics and Statistics  Research Publications
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ItemOn the Connectivity of Visibility GraphsPayne, MS ; Por, A ; Valtr, P ; Wood, DR (SPRINGER, 201210)The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge and vertexconnectivity of visibility graphs. Unless all its vertices are collinear, a visibility graph has diameter at most 2, and so it follows by a result of Plesn\'ik (1975) that its edgeconnectivity equals its minimum degree. We strengthen the result of Plesn\'ik by showing that for any two vertices v and w in a graph of diameter 2, if deg(v) <= deg(w) then there exist deg(v) edgedisjoint vwpaths of length at most 4. Furthermore, we find that in visibility graphs every minimum edge cut is the set of edges incident to a vertex of minimum degree. For vertexconnectivity, we prove that every visibility graph with n vertices and at most l collinear vertices has connectivity at least (n1)/(l1), which is tight. We also prove the qualitatively stronger result that the vertexconnectivity is at least half the minimum degree. Finally, in the case that l=4 we improve this bound to two thirds of the minimum degree.

ItemA LinearTime Algorithm to Find a Separator in a Graph Excluding a MinorReed, B ; Wood, DR (ASSOC COMPUTING MACHINERY, 200910)Let G be an n vertex m edge graph with weighted vertices. A pair of vertex sets A , B ⊆ V ( G ) is a 2/3 separation of order  A ∩ B  if A ∪ B = V ( G ), there is no edge between A − B and B − A , and both A − B and B − A have weight at most 2/3 the total weight of G . Let ℓ ∈ Z + be fixed. Alon et al. [1990] presented an algorithm that in O ( n 1/2 m ) time, outputs either a K ℓ minor of G , or a separation of G of order O ( n 1/2 ). Whether there is a O ( n + m )time algorithm for this theorem was left as an open problem. In this article, we obtain a O ( n + m )time algorithm at the expense of a O ( n 2/3 ) separator. Moreover, our algorithm exhibits a tradeoff between time complexity and the order of the separator. In particular, for any given ϵ ∈ [0,1/2], our algorithm outputs either a K ℓ minor of G , or a separation of G with order O ( n (2−ϵ)/3 in O ( n 1 + ϵ + m ) time. As an application we give a fast approximation algorithm for finding an independent set in a graph with no K ℓminor.

ItemLayout of graphs with bounded treewidthDujmovic, V ; Morin, P ; Wood, DR (SIAM PUBLICATIONS, 2005)

ItemSimultaneous diagonal flips in plane triangulationsBose, P ; Czyzowicz, J ; Gao, Z ; Morin, P ; Wood, DR (WILEY, 200704)

ItemIrreducible triangulations are smallJoret, G ; Wood, DR (ACADEMIC PRESS INC ELSEVIER SCIENCE, 201009)

ItemContractibility and the Hadwiger ConjectureWood, DR (ACADEMIC PRESS LTD ELSEVIER SCIENCE LTD, 201012)

ItemBoundeddegree graphs have arbitrarily large geometric thicknessBarat, J ; Matousek, J ; Wood, D ( 2006)

ItemDrawings of planar graphs with few slopes and segmentsDujmovic, V ; Eppstein, D ; Suderman, M ; Wood, DR (ELSEVIER SCIENCE BV, 200710)

ItemA fixedparameter approach to 2layer planarizationDujmovic, V ; Fellows, M ; Hallett, M ; Kitching, M ; Liotta, G ; McCartin, C ; Nishimura, N ; Ragde, P ; Rosamond, F ; Suderman, M ; Whitesides, S ; Wood, DR (SPRINGER, 200606)

ItemOn the metric dimension of cartesian products of graphsCaceres, J ; Hernando, C ; Mora, M ; Pelayo, IM ; Puertas, ML ; Seara, C ; Wood, DR (SIAM PUBLICATIONS, 2007)