Define a *multiflag graph* to be a triple $G=(F,\mathscr V,\tau)$ consisting of some set $F$, called the set of *flags* of $G$; some subset $\mathscr V\subseteq\mathscr PF$ of the powerset of $F$, called the set of *vertices* of $G$, and a free involution $\tau:F\to F$ on flags. Here, $\mathscr V$ is required to be a *cover* of $F$, that is, $\bigcup\mathscr V=F$.

Every "ordinary" undirected graph $(V,E)$ gives rise to such a structure, with $F$ the incidence relation of the graph, that is, the set of pairs $(v,e)$ where $e\in E$ is an edge and $v\in V$ is a vertex of $e$. Here $\mathscr V$ consists of the sets $F_v:=\{(v,e)\mid\text{$v$ is a vertex of $e$}\}$, one for each $v\in V$ (since $F_v\ne F_{v'}$ for $v\ne v'$, one might identify $V$ with $\mathscr V$). Moreover $\tau(v,e)=(v',e)$ where $v'$ is the (unique) vertex of $e$ different from $v$.

It is well known that in this way one obtains a one-to-one correspondence between undirected graphs and those multiflag graphs for which $\mathscr V$ is not only a cover but also a *partition* of $F$, i.e. satisfies $F_1\cap F_2=\varnothing$ for $F_1\ne F_2$, $F_1,F_2\in\mathscr V$. In this case to each element of $F$ corresponds a unique element of $\mathscr V$ (the one containing it), and one usually pictures flags as "half-edges" $\bullet\!-$, with $\tau$ interchanging two half-edges forming an edge (like ${\bullet\!-}{-\!\bullet}$).

If one drops this additional restriction one can include examples like yours: for

we would have

$F=\{p_x,p_q,q_p,q_v,v_q,v_a,v_w,w_v,w_a,w_x,x_w,x_a,x_p,a_x,a_w,a_v\}$,

$\mathscr V=\{\{p_x,p_q\},\{q_p,q_v\},\{v_q,v_a,v_w\},\{w_v,w_a\},\{w_x,w_a\},\{x_w,x_a,x_p\}\}$

and

$\tau(i_j)=j_i$ for all $i_j\in F$.

This does not give rise to an ordinary graph since $\{w_v,w_a\}\cap\{w_x,w_a\}\ne\varnothing$; one can say that the flag $w_a$ is "two-headed", i. e. describes a "half-edge" with two "heads".

adjacentdifferent toincident? Do you consider only planar graphs? Maybe you meansome of whose \textbf{edges} are like a fork?$\endgroup$flags, together with an equivalence relation (its equivalence classes being vertices) and a free involution (its orbits being edges). If you want something unusual like in this question, you can start with this and then replace equivalence relation with a cover and involutions with something more weird $\endgroup$8more comments