We present a quadratic-time algorithm for computing the genus distribution of any 3-regular 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 edge-amalgamations of some of its subgraphs, in the order corresponding to the post-order traversal of a plane tree that we call the inner tree, and the coordination of that synthesis with just-in-time root-splitting. Submitted:

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We pursue the problem of counting the imbeddings of a graph in each of the orientable surfaces. We demonstrate how to achieve this for an iterated amalgamation of arbitrarily many copies of any graph whose genus distribution is known and further analyzed into a partitioned genus distribution. We introduce the concept of recombinant strands of face-boundary walks, and we develop the use of multiple production rules for deriving simultaneous recurrences. These two ideas are central to a broadbased approach to calculating genus distributions for graphs synthesized from smaller graphs. 1

We investigate the well-known problem of counting graph imbeddings on all oriented surfaces with a focus on graphs that are obtained by pasting together two root-edges of another base graph. We require that the partitioned genus distribution of the base graph with respect to these root-edges be known and that both root-edges have two 2-valent endpoints. We derive general formulas for calculating the genus distributions of graphs that can be obtained either by self-co-amalgamating or by self-contra-amalgamating 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 co-pasted and contra-pasted 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.

We present an O(n 2)-time algorithm for calculating the genus distribution of any 4-regular outerplanar graph. We characterize such graphs in terms of what we call split graphs and incidence trees. The algorithm uses post-order traversal of the incidence tree and productions that are adapted from a previous paper that analyzes double-root vertex-amalgamations and self-amalgamations. 1

We derive a recursion for the genus distribution of the graph family P3✷Pn, with the aid of a modified collection of double-root partials. We introduce a new kind of production, which corresponds to a surgical operation more complicated than the vertex- or edge-amalgamation operations used in our earlier work. 1

Abstract. Star-ladder graphs were introduced by Gross in his development of a quadratic-time algorithm for the genus distribution of a cubic outerplanar graph. This paper derives a formula for the genus distribution of star-ladder 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 star-ladders. 1.

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