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Edge Isoperimetric Problems on Graphs
 Bolyai Math. Series
"... We survey results on edge isoperimetric problems on graphs, present some new results and show some applications of such problems in combinatorics and computer science. 1 Introduction Let G = (V G ; EG ) be a simple connected graph. For a subset A ` VG denote I G (A) = f(u; v) 2 EG j u; v 2 Ag; ` G ..."
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Cited by 16 (5 self)
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We survey results on edge isoperimetric problems on graphs, present some new results and show some applications of such problems in combinatorics and computer science. 1 Introduction Let G = (V G ; EG ) be a simple connected graph. For a subset A ` VG denote I G (A) = f(u; v) 2 EG j u; v 2 Ag; ` G (A) = f(u; v) 2 EG j u 2 A; v 62 Ag: We omit the subscript G if the graph is uniquely defined by the context. By edge isoperimetric problems we mean the problem of estimation of the maximum and minimum of the functions I and ` respectively, taken over all subsets of VG of the same cardinality. The subsets on which the extremal values of I (or `) are attained are called isoperimetric subsets. These problems are discrete analogies of some continuous problems, many of which can be found in the book of P'olya and Szego [99] devoted to continuous isoperimetric inequalities and their numerous applications. Although the continuous isoperimetric problems have a history of thousand years, the dis...
Clusters of Cycles
, 2001
"... A cluster of cycles (or (r; q)polycycle) is a simple planar 2connected finite or countable graph G of girth r and maximal vertexdegree q, which admits an (r; q)polycyclic realization P (G) on the plane. An (r; q)polycyclic realization is determined by the following properties: (i) all interior ..."
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Cited by 2 (2 self)
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A cluster of cycles (or (r; q)polycycle) is a simple planar 2connected finite or countable graph G of girth r and maximal vertexdegree q, which admits an (r; q)polycyclic realization P (G) on the plane. An (r; q)polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q, (ii) all interior faces (denote their number by p r ) are combinatorial rgons, (iii) all vertices, edges and interior faces form a cellcomplex. An example of (r; q)polycycle is the skeleton of (r q ), i.e. of the qvalent partition of the sphere, Euclidean plane or hyperbolic plane by regular rgons. Call spheric pairs (r; q) = (3; 3); (4; 3); (3; 4); (5; 3); (3; 5). Only for those five pairs, P ((r q )) is (r q ) without exterior face; otherwise, P ((r q )) = (r q ). Here we give a compact survey of results on (r; q)polycycles. We start with the following general results for any (r; q)polycycle G: (i) P (G) is unique, except of (easy) case when G is the skeleton of one of 5 Platonic polyhedra; (ii) P (G) admits a cellhomomorphism f into (r q ); (iii) a polynomial criterion to decide if given finite graph is a polycycle, is presented. Call a polycycle proper if it is a partial subgraph of (r q ) and a helicene, otherwise. In [18] all proper spheric polycycles are given. An (r; q)helicene exists if and only if p r ? (q \Gamma 2)(r \Gamma 1) and (r; q) 6= (3; 3). We list the (4; 3), (3; 4)helicenes and the number of (5; 3), (3; 5)helicenes for first interesting p r . Any outerplanar (r; q) polycycle G is a proper (r; 2q \Gamma 2)polycycle and its projection f(P (G)) into (r 2q\Gamma2 ) is convex. Any outerplanar (3; q)polycycle G is a proper (3; q + 2)polycycle. The symmetry group Aut(G) (equal to Aut(P (G)), except ...
Enumeration of generalized polyominoes
"... As a generalization of polyominoes we consider edgetoedge connected nonoverlapping unions of regular kgons. For n ≤ 4 we determine formulas for the number ak(n) of generalized polyominoes consisting of n regular kgons. Additionally we give a table of the numbers ak(n) for small k and n obtained ..."
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Cited by 1 (1 self)
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As a generalization of polyominoes we consider edgetoedge connected nonoverlapping unions of regular kgons. For n ≤ 4 we determine formulas for the number ak(n) of generalized polyominoes consisting of n regular kgons. Additionally we give a table of the numbers ak(n) for small k and n obtained by computer enumeration. We finish with some open problems for kpolyominoes. 1
A Constructive Enumeration of Fusenes and Benzenoids
"... In this paper, a fast and complete method to constructively enumerate fusenes and benzenoids is given. It is fast enough to construct several million non isomorphic structures per second. The central idea is to represent fusenes as labelled inner duals and generate them in a two step approach using ..."
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Cited by 1 (0 self)
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In this paper, a fast and complete method to constructively enumerate fusenes and benzenoids is given. It is fast enough to construct several million non isomorphic structures per second. The central idea is to represent fusenes as labelled inner duals and generate them in a two step approach using the canonical construction path method and the homomorphism principle.
Isoperimetrically Optimal Polyforms
, 2007
"... In the plane, the way to enclose the most area with a given perimeter and to use the shortest perimeter to enclose a given area, is always to use a circle. If we replace the plane by a regular tiling of it, and construct polyforms i.e. shapes as sets of tiles, things become more complicated. We need ..."
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In the plane, the way to enclose the most area with a given perimeter and to use the shortest perimeter to enclose a given area, is always to use a circle. If we replace the plane by a regular tiling of it, and construct polyforms i.e. shapes as sets of tiles, things become more complicated. We need to redefine the area and perimeter measures, and study the consequences carefully. A spiral construction often provides, for every integer number of tiles (area), a shape that is most compact in terms of the perimeter or boundary measure; however it may not exhibit all optimal shapes. We characterize in this paper all shapes that have both shortest boundaries and maximal areas for three common planar discrete spaces. 1
EdgeIsoperimetric Sets of an Infinite Grid
, 2001
"... HBk: N! Ik Let t = d kpme: For (t \Gamma 1)k\Gamma j+1tj\Gamma 1! m ^ (t \Gamma 1)k\Gamma jtj for some 1 ^ j ^ k ..."
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HBk: N! Ik Let t = d kpme: For (t \Gamma 1)k\Gamma j+1tj\Gamma 1! m ^ (t \Gamma 1)k\Gamma jtj for some 1 ^ j ^ k
The square, the triangle and the hexagon
, 1996
"... In this paper we will examine the following problem: What is the minimum number of unit edges required to construct k identical size regular polygons in the plane if sharing of edges is allowed? 1 ..."
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In this paper we will examine the following problem: What is the minimum number of unit edges required to construct k identical size regular polygons in the plane if sharing of edges is allowed? 1
A Constructive Enumeration of Fusenes and Benzenoids
"... In this paper, a fast and complete method to constructively enumerate fusenes and benzenoids is given. It is fast enough to construct several million non isomorphic structures per second. The central idea is to represent fusenes as labelled inner duals and generate them in a two step approach using ..."
Abstract
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In this paper, a fast and complete method to constructively enumerate fusenes and benzenoids is given. It is fast enough to construct several million non isomorphic structures per second. The central idea is to represent fusenes as labelled inner duals and generate them in a two step approach using the canonical construction path method and the homomorphism principle.