A generalized graph in the sense of HRY is a generalization of the notion of a pseudograph given at graph (see in particular Definition in terms action on a set of half edges). By adding structure like directions for the edges, we can define a wheeled graph which is a generalization of a directed pseudograph. What differentiates a generalized graph from a pseudograph and a wheeled graph from a directed pseudograph is the notion of an “exceptional cell,” which should be thought of as a set of half edges which have no vertices.
A generalized graph can be visualized as a set of vertices with half edges attached to them, a rule for attaching half edges to glue the vertices together, and potentially some half edges (flags) that are attached to only one or zero vertices. For instance, a particularly simple generalized graph is one with no vertices and one edge.
The following is Definition 2.2 of HRY:
A generalized graph $G$ is a finite set $Flag(G)$ with:
a partition of $Flag(G)=\coprod_{\alpha\in A} F_\alpha$ into cells with $A$ finite,
a distinguished partition subset $F_\epsilon$ called the exceptional cell,
an involution $\iota$ satisfying $\iota F_\epsilon\subseteq F_\epsilon$,
and a free involution $\pi$ on the set of $\iota$-fixed points of $F_\epsilon$.
Given a generalized graph $G$, Definition 2.3 of HRY gives some useful terminology:
There are a number of important structures that a generalized graph can possess which will be useful in using them to describe . The following is again from HRY:
Suppose $G$ is a generalized graph. Fix a (possibly infinite) set of colors $\mathcal{C}$.
A coloring of $G$ is a function $Flag(G)\overset{\kappa}\to \mathcal{C}$ that is constant on orbits of $\iota$ and $\pi$.
A direction for $G$ is a function $Flag(G)\overset{\delta}\to \{-1,1\}$ such that
For $G$ with direction, an input (resp. output) of a vertex $v$ is a flag $x\in v$ such that $\delta(x)=1$ (resp. $\delta(x)=-1$). An input (resp. output) of $G$ is a leg $x$ such that $\delta(x)=1$ (resp. $\delta(x)=-1$). For $u\in Vt(G)\cup \{G\}$, the set of inputs (resp. outputs) of $u$ is written $in(u)$ (resp. $out(u)$).
A listing for $G$ with direction is a choice for each $u\in Vt(G)\cup \{G\}$ of a bijection of pairs of sets
Thus, using these properties, we can model and with generalized graphs. See there and for more.
, and . Infinity Properads and Infinity Wheeled Properads, Lecture Notes in Mathematics, 2147. Springer, Cham, 2015. (arxiv version)
. Graphs, hypergraphs, and properads, arXiv:1407.3744.
Last revised on July 18, 2016 at 05:26:48. See the history of this page for a list of all contributions to it.