A loop graph is a graph in which loops are allowed: a symmetric, not necessarily irreflexive, relation on a vertex type. Simple graphs are the loopless case, and a loop graph without loops is a simple graph again. A homomorphism of loop graphs sends adjacent vertices to adjacent vertices, so a loop is sent to a loop and an edge to an edge or to a loop, and $\hom(X, Y)$ counts these maps as for simple graphs. Besides homomorphisms and isomorphisms, loop graphs carry here the full complement $\widehat{X}$, which replaces every edge by a non-edge and every loop by a non-loop, the sub-loop-graph induced on a set of vertices, and the edge set, which for a loop graph may contain a pair $vv$, one for each loop.

Loops arise from two constructions on simple graphs. The looped graph $G^\circ$ is obtained from a simple graph $G$ by adding a loop at every vertex; the complement then factors as $\overline{G} = \widehat{G^\circ}$, which is what makes a two-step expansion of homomorphism counts into a complement possible.

The other is a quotient. For a simple graph $F$ and a set $L$ of unordered pairs of vertices, the contraction quotient $F \oslash L$ has as vertices the connected components of the graph on $V(F)$ with edge set $L$, and joins two of them when some edge of $E(F) \setminus L$ joins a vertex of the one to a vertex of the other. It is a graph obtained from $F$ by contracting the edges of $L$ whenever it is loopless; in general it is not, carrying a loop at a component for every edge of $E(F) \setminus L$ with both endpoints inside it, which is why loops must be allowed here.

Two operations on simple graphs accompany these: the spanning subgraph $F_s$, which keeps all vertices of $F$ and those of its edges that lie in a set $s$, and the passage from a set of edges of $F$ to the underlying set of unordered pairs.