nLab semi-graph



A semi-graph is like a graph (more precisely, an undirected, simple graph, possibly with loops) but where certain edges may be ‘open’ or ‘half-open’: they may have a vertex at only one end, or at neither end. As usual, edges can only be joined at vertices. A vertex is typically indicated by a solid dot (possibly with a label), an end of edge without a vertex is typically indicated by a hollow dot (never with a label).

Here is a quick example, which has one edge without any vertices, and two edges with only one vertex.


Throughout this page, we pick a set with exactly one element, and denote it 11.


The following definition follows Mochizuki2006.


A semi-graph consists of a set VV, a set EE whose elements are sets with exactly two elements, and, for every element ee of EE, a map ζ e:eV1\zeta_{e}: e \rightarrow V \sqcup 1.


Let GG be a semi-graph. A vertex of GG is an element of VV. An edge of GG is an element of EE. An edge ee of GG abuts to a vertex vv of GG if vv belongs to the image of ζ e\zeta_{e}. We refer to an element of an edge ee of GG as a branch of ee.


In particular, every edge of a semi-graph has exactly two branches, but it is possible that neither, both, or only one of the branches abuts a vertex. Thus we think of an edge ee as subdivided into two branches b 1b_{1} and b 2b_{2}, and do not think of the meeting point as a vertex of the graph; but always consider the other end of each branch as a vertex, either ‘closed’ (the branch maps to an actual vertex of GG under ζ e\zeta_{e}) or ‘open’ (the branch maps to 11 under ζ e\zeta_{e}).


Last revised on April 17, 2020 at 06:24:09. See the history of this page for a list of all contributions to it.