边剖分、点扩张与图的最大亏格的可约性

Subdivision of An Edge, Extension of a Vertex and Reducibility of Maximum Genus of a Graph

设γM（G）是连通图G=（V,E）的最大亏格,记EM-（G）={e ∈ E（G）G\e连通,且γM（G\e）=γM（G）}.若EM-（G）≠ ,则称G是γM（G）-可约的;否则称G是γM（G）-不可约的.本文证明了边的剖分不改变图的最大亏格可约性,点的扩张不改变上可嵌入图的最大亏格可约性;并给出了两类满足EM-（G）=E（G）的非4-边连通图.

Let γM (G) be the maximum genus of a connected graph G = (V, E), EM-(G) = {e∈E(G)\G\e is connected and γM(G\e)=γM(G)}. G is γM(G)-reducible if EM-(G)≠ ; otherwise G is γM(G)-irreducible. In this paper we show that subdivision of an edge does not change the reducibility of maximum genus of a graph and extension of a vertex does not change the reducibility of maximum genus of up-embeddable graphs; Two classes of non-4-edge connected graphs satisfying EM-(G) = E(G) are given.