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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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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.
kMinimal Triangulations of Surfaces
, 1995
"... A triangulation of a closed surface is kminimal (k 3) if each edge belongs to some essential kcycle and all essential cycles have length at least k. It is proved that the class of kminimal triangulations is finite (up to homeomorphism). As a consequence it follows, without referring to the R ..."
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A triangulation of a closed surface is kminimal (k 3) if each edge belongs to some essential kcycle and all essential cycles have length at least k. It is proved that the class of kminimal triangulations is finite (up to homeomorphism). As a consequence it follows, without referring to the RobertsonSeymour's theory, that there are only finitely many minorminimal graph embeddings of given representativity. In the topological part, certain separation properties of homotopic simple closed curves are presented.
Irreducible Triangulations of Surfaces with Boundary
, 2011
"... A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly nonorientab ..."
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A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly nonorientable) surface of genus g ≥ 0 with b ≥ 0 boundaries is O(g +b). So far, the result was known only for surfaces without boundary (b = 0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.
Graphs on Surfaces and the Partition Function of String Theory
, 2008
"... Abstract. Graphs on surfaces is an active topic of pure mathematics belonging to graph theory. It has also been applied to physics and relates discrete and continuous mathematics. In this paper we present a formal mathematical description of the relation between graph theory and the mathematical phy ..."
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Abstract. Graphs on surfaces is an active topic of pure mathematics belonging to graph theory. It has also been applied to physics and relates discrete and continuous mathematics. In this paper we present a formal mathematical description of the relation between graph theory and the mathematical physics of discrete string theory. In this description we present problems of the combinatorial world of real importance for graph theorists. The mathematical details of the paper are as follows: There is a combinatorial description of the partition function of bosonic string theory. In this combinatorial description the string world sheet is thought as simplicial and it is considered as a combinatorial graph. It can also be said that we have embeddings of graphs in closed surfaces. The discrete partition function which results from this procedure gives a sum over triangulations of closed surfaces. This is known as the vacuum partition function. The precise calculation of the partition function depends on combinatorial calculations involving counting all nonisomorphic triangulations and all spanning trees of a graph. For this reason the exact computation of the partition function turns out to be very complicated, however we show the exact expressions for its computation for the case of any closed orientable surface. We present as specific cases a clear computation for the sphere and the way it is done for the torus, and for the nonorientable case of the projective plane. 1