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Genus distribution of graph amalgamations: selfpasting at rootvertices
 AUSTRALASIAN JOURNAL OF COMBINATORICS VOLUME 49 (2011), PAGES 19–38
, 2011
"... Counting the number of imbeddings in various surfaces of each of the graphs in an interesting family is an ongoing topic in topological graph theory. Our special focus here is on a family of closed chains of copies of a given graph. We derive doubleroot partials for open chains of copies of a given ..."
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Cited by 11 (11 self)
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Counting the number of imbeddings in various surfaces of each of the graphs in an interesting family is an ongoing topic in topological graph theory. Our special focus here is on a family of closed chains of copies of a given graph. We derive doubleroot partials for open chains of copies of a given graph, and we then apply a selfamalgamation theorem, thereby obtaining genus distributions for a sequence of closed chains of copies of that graph. We use recombinant strands of faceboundary walks, and we further develop the use of multiple production rules in deriving partitioned genus distributions.
Genus Distribution of Graph Amalgamations: Pasting When One Root Has Arbitrary Degree
 ARS MATHEMATICA CONTEMPORANEA
"... This paper concerns counting the imbeddings of a graph in a surface. In the first installment of our current work, we showed how to calculate the genus distribution of an iterated amalgamation of copies of a graph whose genus distribution is already known and is further analyzed into a partitioned g ..."
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Cited by 9 (9 self)
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This paper concerns counting the imbeddings of a graph in a surface. In the first installment of our current work, we showed how to calculate the genus distribution of an iterated amalgamation of copies of a graph whose genus distribution is already known and is further analyzed into a partitioned genus distribution (which is defined for a doublerooted graph). Our methods were restricted there to the case with two 2valent roots. In this sequel we substantially extend the method in order to allow one of the two roots to have arbitrarily high valence.
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 8 (8 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.
Genus distribution of 4regular outerplanar graphs
, 2011
"... We present an O(n 2)time algorithm for calculating the genus distribution of any 4regular outerplanar graph. We characterize such graphs in terms of what we call split graphs and incidence trees. The algorithm uses postorder traversal of the incidence tree and productions that are adapted from a ..."
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Cited by 7 (7 self)
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We present an O(n 2)time algorithm for calculating the genus distribution of any 4regular outerplanar graph. We characterize such graphs in terms of what we call split graphs and incidence trees. The algorithm uses postorder traversal of the incidence tree and productions that are adapted from a previous paper that analyzes doubleroot vertexamalgamations and selfamalgamations.
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 4 (4 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 3 (3 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.
Lower Bounds for the Average Genus of a CFgraph
"... CFgraphs form a class of multigraphs that contains all simple graphs. We prove a lower bound for the average genus of a CFgraph which is a linear function of its Betti number. A lower bound for average genus in terms of the maximum genus and some structure theorems for graphs with a given average ..."
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Cited by 1 (0 self)
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CFgraphs form a class of multigraphs that contains all simple graphs. We prove a lower bound for the average genus of a CFgraph which is a linear function of its Betti number. A lower bound for average genus in terms of the maximum genus and some structure theorems for graphs with a given average genus are also provided. 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
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.