Combinatorics on Adjacency Graphs and Incidence

Computer Science Department of The University of Auckland CITR at Tamaki Campus (http://www.citr.auckland.ac.nz/)

CITR-TR-121

Jan 2003

Combinatorics on Adjacency Graphs and Incidence Pseudographs Reinhard Klette 1

Abstract The paper starts with a brief review of combinatorial results for adjacency and oriented adjacency graphs (combinatorial maps). The main subject is incidence pseudographs (a dual of cell complexes) of the n-dimensional orthogonal grid. It is well known that these pseudographs (or complexes) allow a definition of a topological space, and combinatorial formulas are provided for characterizing open and closed sets in this topology. The paper extends work by K. Voss in 1993, which is (in the terminology of incidence pseudographs) on open regions only. The paper also provides combinatorial formulas for closed regions. Matching formulas and Euler characteristic calculations are generalized for arbitrary open or closed regions in the n-dimensional orthogonal grid.

1

[email protected] CITR Tamaki The University of Auckland, Tamaki Campus Morrin Road, Glen Innes, Auckland 1005, New Zealand

You are granted permission for the non-commercial reproduction, distribution, display, and performance of this technical report in any format, BUT this permission is only for a period of 45 (forty-five) days from the most recent time that you verified that this technical report is still available from the CITR Tamaki web site under terms that include this permission. All other rights are reserved by the author(s).


  

  "!$#%'&()  *+ ,-. 0/1.2 -
)34%5&  6 7,8:9=@?BA-CDE7FBFG7 Hb,dfIKJMeLNgWhiJPS@j\ORO,QYlkT JPO QnO,S@T>hoO QqpZgl^  J W U  V  X Z Y \ T R [ B V G ] _ ^ \ T _ ` c a rsb]i]iTtY Lb O kum>vMj\VBY-IEYZYlVo^omWegWhiS@j\O,Ylk5wox xzy{ml|"Vo}~WVoOj\O,Ylk €fƒ‚l„@… †@†‡…@ˆ@‰,Š‡‹,‚‡„@‰,Œzf_‰@‹PEŒ‡Ž KdcQb]O’TtO ‘oYs“k”oœi^_g‡O •‡h›™Z–@VB“ œ¡YWVo—hoh›`aJT\UlO^V4YWTtkcYWplhBO,b,T\p˜k‡]iVBT\VBVB]YlY‡ho^i`i`iVV›O,kspW]G`i^iO ^VBkgl}4œik‡O T\b ho`_U.VG,YW]GOšOhopZac™‡Ul]i,^T\]GVBOd’žKOqpZ]GUlV›k{[ ^glT\VoO,žKho}Nj¢b b,Qb,dPd>™Zh›hoTtVBYlbRjtj>O Qš`iho™‡bb Tt]iQYWT\O,ORp‡j`ij\b,VoQ]i£@T\O,VoOpl^Gj’Ÿ^G]GŸBb,Vo ‡d¢^_Jg‡`_j\UWUW`iVV^ ¤ZžKb,¥ ]4k{T\hoQb QVGYWp‡^_j\T\Vob£@YWVoO^GjfŸ"b,O]Gjt`_j\Ulb}¦b  bOYWk‡OVBj>§‡,YZ]iT\T\`_kuT\ ubY5Iƒ`Pb,T\^d¢}OVG`ijtb,jg‡j\J4O Ul^VO]GpWVMOp‡pu]GVBb,]q[ VoT\k‡£@Vo`iksVBYldKk‡b,^]}hiUWb,O]i]GScO ™‡ho`iaVB]iªT\¨G«TtYW¬lc­ b,®Bpu®PVBTtY¯Y©O,woYl°R° k)±ZhBmuj\}bR^iU‡VoT\k)h²U ^iV›T\`i^^+žETtTtYsY-`_`_U‡UWT\V^ `iO,VBj\^i]Gb QTtpZYl]Gb,bj\[ bRT\ k‡aVo^)b,d>hoTtbRYWQšhGT\™‡kZTtVBYlYlO ho`iV"b,]iplT\O^ijVBgld³kZb,bR]GQ,]GgZO,j\p‡ORUl^.^GŸMd³b,b,]YchBbj\bRpu^iVBVoYk´]G]GVoVo,T\bT\b,YWYl^^ob YZrsj\a O  ‡`iJhiUZUlTtV4YlplO,dKp˜b,VB] ] Qb,p˜gZVBj\YsOR^"bO,]"YlhGk)j\b µP^iVog‡ksj\VB]G]Vo,hiUWT\bOYW]G^O hoTtY `iVG]i`_T\UW^iV`_T\hhok@O,T\j\QhGgZVBj\YlOR^_`_T\T\b,b,YlYuO,^qjO,b,]G]GV`_UlRb VB YWbVGYW]GO,OjtjT\¨oVo]iksT\k