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 double-root partials for open chains of copies of a given graph, and we then apply a self-amalgamation theorem, thereby obtaining genus distributions for a sequence of closed chains of copies of that graph. We use recombinant strands of face-boundary walks, and we further develop the use of multiple production rules in deriving partitioned genus distributions.

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 double-rooted graph). Our methods were restricted there to the case with two 2-valent roots. In this sequel we substantially extend the method in order to allow one of the two roots to have arbitrarily high valence.

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 an O(n2)-time algorithm for calculating the sequence of numbers g0(G), g1(G), g2(G),... of distinct ways to embed a 3-regular 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 recombinant-strands reassembly process that fits pieces together three-at-a-vertex. Key algorithmic features are reassembly along a post-order traversal, with just-in-time dynamic assignment of roots for quadrangular pieces encountered along the tour. 1.

CF-graphs form a class of multigraphs that contains all simple graphs. We prove a lower bound for the average genus of a CF-graph 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

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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