Reconstructibility versus edge reconstructibility of infinite graphs

✍ Scribed by Carsten Thomassen

• Publisher: Elsevier Science
• Year: 1978
• Tongue: English
• Weight: 151 KB
• Volume: 24
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(78)90203-0

✦ Synopsis

RECONgTRUCTIBILITY VERSUI~ EDGE

RECONSTR1UCTIBILtlY OF !NF![?CTE GN~APNS Cars,~en -FI-!Ob,~ ASSEN A.hah,,*~atL~k /~.t;tir~*., t h~ieersi;e~sp ~tk~'n, S0{P} Aarbus C. Detm~a& Rcc~ .d 23 [;cccm~cr 1~)77 [~¢ :{>.cd 7 April D)TS

For every cm~dma! a >R o ~here exi::ts an ,:t-rQ,',ular .g;api~ w[?icll is R "msm~.ctible btl~. not e : reconst'uctible. indums an isomorphism * _+ ~r :H(a, k) H~a, m) which implies, by [3], that k = m. Also it follow~ from the discuss{o~ of ~] that for every edge e of g(a, .8. k) L(a, 13. k)--e:--~L(o',~,,,n) for:;ome ,~ ~k--2.

and that for each m ~max (k-! 0)

Ua, 13, k)-e-'--L(a,/3, n~)

for ~ edges e. Thus L(a./3, l) and Li~, {3,0) h~ve the same fami!.{es of edgedeleted subgraphs but are non-is~:morpn;c. In particulac. G '~'~ = L(~,/3, 0 is nm edge reconstructible. In order to complete th:~ proof we pro~e ~.ha~ f>~ is reconste,~cfibie. Fr:~m the discussion of [3J it folkm~; that R.r any vezI:ex x in G',,x, k)-B(m k).