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Genus distributions of cubic outerplanar graphs
, 2011
"... We present a quadratictime algorithm for computing the genus distribution of any 3regular outerplanar graph. Although recursions and some formulas for genus distributions have previously been calculated for bouquets and for various kinds of ladders and other special families of graphs, cubic outer ..."
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Cited by 15 (15 self)
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We present a quadratictime algorithm for computing the genus distribution of any 3regular outerplanar graph. Although recursions and some formulas for genus distributions have previously been calculated for bouquets and for various kinds of ladders and other special families of graphs, cubic outerplanar graphs now emerge as the most general family of graphs whose genus distributions are known to be computable in polynomial time. The key algorithmic features are the syntheses of the given outerplanar graph by a sequence of edgeamalgamations of some of its subgraphs, in the order corresponding to the postorder traversal of a plane tree that we call the inner tree, and the coordination of that synthesis with justintime rootsplitting.
Genus distributions of graphs under selfedgeamalgamations
 Ars Math. Contemporanea
"... We investigate the wellknown problem of counting graph imbeddings on all oriented surfaces with a focus on graphs that are obtained by pasting together two rootedges of another base graph. We require that the partitioned genus distribution of the base graph with respect to these rootedges be know ..."
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Cited by 10 (10 self)
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We investigate the wellknown problem of counting graph imbeddings on all oriented surfaces with a focus on graphs that are obtained by pasting together two rootedges of another base graph. We require that the partitioned genus distribution of the base graph with respect to these rootedges be known and that both rootedges have two 2valent endpoints. We derive general formulas for calculating the genus distributions of graphs that can be obtained either by selfcoamalgamating or by selfcontraamalgamating a base graph whose partitioned genus distribution is already known. We see how these general formulas provide a unified approach to calculating genus distributions of many new graph families, such as copasted and contrapasted closed chains of copies of the triangular prism graph, as well as graph families like circular and Möbius ladders with previously known solutions to the genus distribution problem.
Embeddings of Graphs of Fixed Treewidth and Bounded Degree
, 2013
"... Let F be any family of graphs of fixed treewidth and bounded degree. We construct a quadratictime algorithm for calculating the genus distribution of the graphs in F. Within a postorder traversal of the decomposition tree, the algorithm involves a fullpowered upgrading of production rules and roo ..."
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Cited by 7 (7 self)
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Let F be any family of graphs of fixed treewidth and bounded degree. We construct a quadratictime algorithm for calculating the genus distribution of the graphs in F. Within a postorder traversal of the decomposition tree, the algorithm involves a fullpowered upgrading of production rules and rootpopping. This algorithm for calculating genus distributions in quadratic time complements an algorithm of Kawarabayashi, Mohar, and Reed for calculating the minimum genus of a graph of bounded treewidth in linear time.
Embeddings of cubic halin graphs: a surfacebysurface inventory
 Ars Mathematica Contemporanea
"... We derive an O(n2)time algorithm for calculating the sequence of numbers g0(G), g1(G), g2(G),... of distinct ways to embed a 3regular Halin graph G on the respective orientable surfaces S0, S1, S2,.... Key topological features are a quadrangular decomposition of plane Halin graphs and a new recomb ..."
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Cited by 4 (4 self)
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We derive an O(n2)time algorithm for calculating the sequence of numbers g0(G), g1(G), g2(G),... of distinct ways to embed a 3regular Halin graph G on the respective orientable surfaces S0, S1, S2,.... Key topological features are a quadrangular decomposition of plane Halin graphs and a new recombinantstrands reassembly process that fits pieces together threeatavertex. Key algorithmic features are reassembly along a postorder traversal, with justintime dynamic assignment of roots for quadrangular pieces encountered along the tour. 1.
GENUS DISTRIBUTIONS OF STARLADDERS
"... Abstract. Starladder graphs were introduced by Gross in his development of a quadratictime algorithm for the genus distribution of a cubic outerplanar graph. This paper derives a formula for the genus distribution of starladder graphs, using Mohar’s overlap matrix and Chebyshev polynomials. Newly ..."
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Abstract. Starladder graphs were introduced by Gross in his development of a quadratictime algorithm for the genus distribution of a cubic outerplanar graph. This paper derives a formula for the genus distribution of starladder graphs, using Mohar’s overlap matrix and Chebyshev polynomials. Newly developed methods have led to a number of recent papers that derive genus distributions and total embedding distributions for various families of graphs. Our focus here is on a family of graphs called starladders. 1.
ENUMERATION OF DIGRAPH EMBEDDINGS
, 2013
"... Abstract. A cellular embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. The directed genus polynomial of an Eulerian digraph D is the polynomial ΓD(x) = h≥0 gh(D)x h where gh(D) is the number of directed embedd ..."
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Cited by 1 (0 self)
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Abstract. A cellular embedding of an Eulerian digraph D into a closed surface is said to be directed if the boundary of each face is a directed closed walk in D. The directed genus polynomial of an Eulerian digraph D is the polynomial ΓD(x) = h≥0 gh(D)x h where gh(D) is the number of directed embeddings into the orientable surface Sh, of genus h, for h = 0, 1,.... The sequence {gh(D)h ≥ 0}, which is called the directed genus distribution of the digraph D, is known for very few classes of graphs, compared to the genus distribution of a graph. This paper introduces a variety of methods for calculating the directed genus distributions of Eulerian digraphs. We use them to derive an explicit formula for the directed genus distribution of any 4regular outerplanar digraph. We show that the directed genus distribution of such a digraph is determined by the redblue star decompositions of the characteristic tree for an outerplanar embedding. The directed genus distribution of a 4regular outerplanar digraph is proved to be logconcave, which is consistent with an affirmative answer to a question of Bonnington, et al. [2]. Indeed, the corresponding genus polynomial is realrooted. We introduce Eulerian splitting at a vertex of a digraph, and we prove a splitting theorem for digraph embedding distributions that is analogous to the splitting theorem for (undirected) graph embedding distributions. This new splitting theorem allows conversion of the enumeration of embeddings of a digraph with vertex degrees larger than 4 into a problem of enumerating the embeddings of some 4regular digraphs. 1.
GENUS DISTRIBUTIONS OF CUBIC SERIESPARALLEL GRAPHS
"... Abstract. We derive a quadratictime algorithm for the genus distribution of any 3regular, biconnected seriesparallel graph, which we extend to any biconnected seriesparallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are seriesparallel gra ..."
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Abstract. We derive a quadratictime algorithm for the genus distribution of any 3regular, biconnected seriesparallel graph, which we extend to any biconnected seriesparallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are seriesparallel graphs, this yields, by use of baramalgamation, a quadratictime algorithm for every graph of treewidth at most 2 and maximum degree at most 3. Version: 08:11 April 13, 2014 1.
Genus Distribution of P3✷Pn
"... We derive a recursion for the genus distribution of the graph family P3✷Pn, with the aid of a modified collection of doubleroot partials. We introduce a new kind of production, which corresponds to a surgical operation more complicated than the vertex or edgeamalgamation operations used in our ea ..."
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We derive a recursion for the genus distribution of the graph family P3✷Pn, with the aid of a modified collection of doubleroot partials. We introduce a new kind of production, which corresponds to a surgical operation more complicated than the vertex or edgeamalgamation operations used in our earlier work. 1