Remodeling a DS-diagram into one with E-cycle
Hiroshi Ikeda, Masakatsu YAMASHITA, Kazuo Yokoyama · Tokyo Journal of Mathematics · 2000
directly whether a standard spine is a flow-spine or not.By DS-diagrams (see Definition 1.1), we get rid of the difficulty.It is known in [5] that any closed 3-manifold has a DS-diagram constructed from a standard spine.The flow-spines correspond to the DS-diagrams with E- cycle, see [4] and [8].Thus the problem above can be translated into the remodeling problem of a DS-diagram into one with E-cycle (see Definition 2.2).The main theorem of this paper can be stated as follows (see Definition 1.2 for the notion of DS-isomorphism).THEOREM 1.1.Any DS-diagram is DS-isomorphic to a DS-diagram with E-cycle.We prove this theorem by finding a DS-isomorphism to get a DS-diagram with E-cycle algorithmically.Including the concept of DS-isomorphism, let us review briefly some of the definitions made in [4] through [8] to understand the theorem.Consider a 2-sphere $S^{2}$ and a connected 3-regular graph $G$ embedded in $S^{2}$ .Let $V_{G}$ be the set of vertices of $G$ .Then $G$ induces a natural structure of cell complex $K(G)$ on $S^{2}$ ; O-cells are elements of $V_{G}$ , l-cells are the connected components of $G-V_{G}$ and 2-cells are the connected components of $S^{2}-G$ .For a definition of cell complexes, see for example, [9].DEFINITION 1.1.A triple $\Delta=(S^{2}, G, f)$ is called a DS-diagram if