# nLab signed graph

A signed graph is an undirected simple graph (a set $V$ of vertices together with a collection $E$ of two-element subsets of $V$ called edges) plus a function from $E$ to a two-element set of “signs”, e.g., $\sigma: E \to \{+, -\}$. A morphism of signed graphs is a function between vertex sets which preserves signed graph structure.

If $\binom{V}{2}$ denotes the set of two-element subsets of $V$, a signed graph could be equivalently defined to be a set $V$ together with a partial function $\binom{V}{2} \to \{+, -\}$. The domain of this partial function is declared to be $E$, so the data of a signed graph can be represented as $V$ together with a span of type

$\binom{V}{2} \hookleftarrow E \stackrel{\sigma}{\to} \{+, -\}.$

Signed graphs should not be regarded as directed graphs: signs on undirected edges are not the same as directional information. For example, if vertices represent people and edges represent being acquainted then the signs could indicate being allies or enemies.

If a linear order has been imposed on the vertex set, then the sign function can be interpreted as imparting directions to edges (say if $x \lt y$ in the order and an edge $e$ between $x$ and $y$ has sign $+$, then the direction is $x \to y$; if $e$ has sign $-$, then the direction is $y \to x$). However, such linear orders on vertices will not be preserved by morphisms of signed graphs, so this does not give a functorial way to derive directed graphs from signed graphs.

Signed graphs (and the basic theorems of their theory) seem to be rediscovered again and again. See Zaslavsky for an annotated bibliography on signed graphs and their various guises.