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Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Cited by 33 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
Algebraic shifting and fvector theory
, 2007
"... This manuscript focusses on algebraic shifting and its applications to fvector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the gconjecture for simplicial spheres, which is considered by many as the main open problem in f ..."
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Cited by 3 (1 self)
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This manuscript focusses on algebraic shifting and its applications to fvector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the gconjecture for simplicial spheres, which is considered by many as the main open problem in fvector theory. While this goal has not been achieved, related results of independent interest were obtained, and are presented here. The operator algebraic shifting was introduced by Kalai over 20 years ago, with applications mainly in fvector theory. Since then, connections and applications of this operator to other areas of mathematics, like algebraic topology and combinatorics, have been found by different researchers. See Kalai’s recent survey [34]. We try to find (with partial success) relations between algebraic shifting and the following other areas: • Topological constructions on simplicial complexes. • Embeddability of simplicial complexes: into spheres and other manifolds.
FourTerminal Reducibility and ProjectivePlanar WyeDeltaWyeReducible Graphs
 J. GRAPH THEORY
, 2000
"... A graph is Y∆Y reducible if it can be reduced to a vertex by a sequence of seriesparallel reductions and Y∆Ytransformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that fourterminal planar graphs are Y∆Yreducible w ..."
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Cited by 2 (0 self)
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A graph is Y∆Y reducible if it can be reduced to a vertex by a sequence of seriesparallel reductions and Y∆Ytransformations. Terminals are distinguished vertices which cannot be deleted by reductions and transformations. In this paper we show that fourterminal planar graphs are Y∆Yreducible when at least three of the vertices lie on the same face. Using this result we characterize Y∆Yreducible projectiveplanar graphs. We also consider terminals in projectiveplanar graphs, and establish that graphs of crossingnumber one are Y∆Yreducible.
Topological Graph Theory  A Survey
 Cong. Num
, 1996
"... this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1 ..."
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Cited by 1 (0 self)
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this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1
On Embeddability and Stresses of Graphs
, 2007
"... Gluck [6] has proven that triangulated 2spheres are generically 3rigid. Equivalently, planar graphs are generically 3stress free. We show that already the K5minor freeness guarantees the stress freeness. More generally, we prove that every Kr+2minor free graph is generically rstress free for 1 ..."
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Cited by 1 (0 self)
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Gluck [6] has proven that triangulated 2spheres are generically 3rigid. Equivalently, planar graphs are generically 3stress free. We show that already the K5minor freeness guarantees the stress freeness. More generally, we prove that every Kr+2minor free graph is generically rstress free for 1 ≤ r ≤ 4. (This assertion is false for r ≥ 6.) Some further extensions are discussed. 1 Introduction and
”Doctor of Philosophy”
"... This manuscript focusses on algebraic shifting and its applications to fvector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the gconjecture for simplicial spheres, which is considered by many as the main open problem in f ..."
Abstract
 Add to MetaCart
This manuscript focusses on algebraic shifting and its applications to fvector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the gconjecture for simplicial spheres, which is considered by many as the main open problem in fvector theory. While this goal has not been achieved, related results of independent interest were obtained, and are presented here. The operator algebraic shifting was introduced by Kalai over 20 years ago, with applications mainly in fvector theory. Since then, connections and applications of this operator to other areas of mathematics, like algebraic topology and combinatorics, have been found by different researchers. See Kalai’s recent survey [34]. We try to find (with partial success) relations between algebraic shifting and the following other areas: • Topological constructions on simplicial complexes. • Embeddability of simplicial complexes: into spheres and other manifolds.