Abstract
The above authors [2] and S. Stahl [3] have shown that if a graphG is the 2-amalgamation of subgraphsG 1 andG 2 (namely ifG=G 1∪G 2 andG 1∩G 2={x, y}, two distinct points) then the orientable genus ofG,γ(G), is given byγ(G)=γ(G 1)+γ(G 2)+ε, whereε=0,1 or −1. In this paper we sharpen that result by giving a means by whichε may be computed exactly. This result is then used to give two irreducible graphs for each orientable surface.
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References
S. Alpert, The genera of amalgamations of graphs,Trans. Am. Math. Soc. 178 (1973), 1–39.
R. Decker, H. Glover andJ. P. Huneke, The genus of the 2-amalgamations of graphs,Journal of Graph Theory 5 (1981), 95–102.
S. Stahl, Permutation-partition pairs: a combinatorial generalization of graph embedding.Trans. Am. Math. Soc. 259 (1980), 129–145.