@techreport{TR-IC-25-07,
   number = {IC-25-07},
   author = {L. G. S. {Gonzaga} and C. N. Campos},
   title = {{Improper colourings of outerplanar graphs}},
   month = {December},
   year = {2025},
   institution = {Institute of Computing, University of Campinas},
   note = {In English, 12 pages.
    \par\selectlanguage{english}\textbf{Abstract}
       A  $(k,d)$-colouring  of a graph $G$ is a vertex colouring with
       at  most  $k$  colours  such that every monochromatic connected
       component   has   maximum   degree   at   most   $d$.  Thus,  a 
       $(k,0)$-colouring  corresponds to a proper $k$-colouring. It is
       well  known  that every planar graph admits $(3,2)$-colourings,
       but  there  is  no  $d$  for  which  every  planar graph admits
       $(1,d)$-colourings  or  $(2,d)$-colourings.  Moreover, deciding
       whether   a   planar   graph   admits  a  $(2,1)$-colouring  is 
       NP-complete.  It  is  also  known  that every outerplanar graph
       admits    $(3,0)$   and   $(2,2)$-colourings,   although   some  
       outerplanar   graphs  do  not  admit  $(2,1)$-colourings,  with 
       triangles  playing  an  important  role. In this work, we prove
       that   deciding   the   existence   of   $(2,1)$-colourings  of 
       apart-plane  graphs  --  in  which  every  pair of triangles is
       disjoint  --  is  also  NP-complete.  Moreover,  we  present  a 
       polynomial-time  algorithm  for  deciding  whether an outerpath
       graph  --  an  outerplane  graph  for  which  the  weak dual is
       isomorphic  to  a path -- admits $(2,1)$-colourings. This fully
       characterises the $(2,1)$-colourability of outerpath graphs and
       provides  new insights into the structural role of triangles in
       defective colouring problems.
  }
}