Recursive formulas for embedding distributions of cubic outerplanar graphs

Recently, the first author and his coauthor proved a k-order homogeneous linear recursion for the genus polynomials of any H-linear family of graphs (called path-like graph families by Mohar). Cubic outerplanar graphs are tree-like graph families. In this paper, we derive a recursive formula for the total embedding distribution of any cubic outerplanar graph. We also obtain explicit formulas for the number of embeddings of cubic outerplanar graphs into the plane, torus, projective plane and Klein bottle. In addition, we present a O (n(h +Δ))-time algorithm to compute the genus distribution and the crosscap number distribution of any cubic outerplanar graph, where h and Δ are the height and maximum degree of the characteristic tree, respectively. We have written an efficient enumeration program in C++ for computing this recursive function and constructing tables of genus distributions of cubic outerplanar graphs. Our program is documented and available on request. ∗ The work of the first author was supported by National Natural Science Foundation of China (Grant No. 11471106) and NSFC of Hunan (Grant No. 14JJ2043). † Corresponding author. The work of the second author was supported by NNSFC under Grant No. 61673048. Y. CHEN AND T. WANG/AUSTRALAS. J. COMBIN. 68 (1) (2017), 131–146 132