By Ibrahim Assem
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Additional resources for Algèbres et Modules
Moreover, the two vertices can be combined into a single vertex with e deleted, and the resulting graph is still simply connected. Such combination of two vertices into one is how inner coface maps are defined in the finite ordinal category and the Moerdijk-Weiss dendroidal category . The obvious analog is not true for connected wheel-free graphs in general. 16) with at least two ordinary edges, if the two vertices are combined with some but not all of the ordinary edges deleted, then the resulting graph has a directed loop at the combined vertex.
Therefore, once v is deleted, the resulting graph cannot be connected. This contradicts the assumption on v, so there is only one ordinary edge adjacent to v. 2 Extremal Paths Later we will need to know that a connected wheel-free graph with at least two vertices always has at least two almost isolated vertices. 16), both vertices are almost isolated. When there are more than two vertices in a connected wheel-free graph, we need the following concept to show the existence of almost isolated vertices.
Such a path is said to have length r. 2. A path of length 0 is called a trivial path. A path of length 1 is called an internal path. 3. , terminal vertex). An end vertex means either an initial vertex or a terminal vertex. 4. An internal path whose initial vertex is equal to its terminal vertex is called a cycle. Otherwise, it is called a trail. 5. A directed path in G is an internal path P as above such that each ej has initial vertex vj 1 and terminal vertex vj . 6. A wheel in G is a directed path that is also a cycle.
Algèbres et Modules by Ibrahim Assem