insertion leads then to $\Delta(L)=2$. This is the case, for example, in the mass mechanism and in the $W_R-N-W_R$ diagram of LR-symmetric models. The other possibility is to have LNV vertices. For the cases with a doubly charged bilepton $S'(+2, {\bf 1})$, for example, one can have one $\Delta(L)=2$ vertex. And, finally, it is possible to have models with two $\Delta(L)=1$ vertices. An example is trilinear RPV SUSY, e.g., from the superpotential $\mathcal{W} = \lambda' {\widehat L}{\widehat Q}{\widehat D}^c$.

Most of the new models we find are of the short-range type and thus should be testable at the LHC. We will discuss one particular example in greater detail in Sec. \[sec:example\].

Decomposition of topology II
----------------------------

  ---- --------------------------------------------- ----------------------------- ----------------------------- ----------------------------- -----------------------------------------------------
                                                                                                                                               
   \#                  Decomposition                       $S$ or $V_{\rho}$            $S'$ or $V'_{\rho}$         $S'' $ or $V''_{\rho}$     Models/Refs./Comments
   1    $(\bar{u} d) (\bar{u} d) (\bar{e} \bar{e})$         $ (+1,{\bf 1})$               $ (+1,{\bf 1})$              $ (-2, {\bf 1})$        Addl. triplet scalar [@Mohapatra:1981pm]
                                                                                                                                               LR-symmetric models [@Rizzo:1982kn; @Hirsch:1996qw]
                                                            $ (+1,{\bf 8})$               $ (+1,{\bf 8})$              $ (-2, {\bf 1})$        
   2    $(\bar{u} d) (\bar{u} \bar{e}) (\bar{e} d)$         $(+1,{\bf 1})$               $(-1/3,{\bf 3})$          $(-2/3,\overline{\bf 3})$   
                                                            $(+1,{\bf 8})$               $(-1/3,{\bf 3})$          $(-2/3,\overline{\bf 3})$   
   3    $(\bar{u} \bar{u}) (dd) (\bar{e} \bar{e})$     $(+4/3,\overline{\bf 3})$         $(+2/3,{\bf 3})$               $(-2,{\bf 