ON RECONSTRUCTING DISCONNECTED GRAPHS

Gary Char t rand , Western Michigan Un ive r s i ty Hudson V. Kronk, SUNY a t Binghamton

The Recons t ruc t ion Con jec tu re states t h a t any graph G of o r d e r a t l e a s t t h r e e can b e determined uniquely o r is r e c o n s t r u c t a b l e from its proper maximal induced subgraphs G-v, v E V(G). A number of proper- t ies of G can be e a s i l y de te rmined from t h e s e subgraphs , among which a r e t h e number of v e r t i c e s , t h e number of edges , i t s degree sequence , and whether i t i s connected ( o r , e q u i v a l e n t l y , whether i t is discon- n e c t e d ) . It i s a l s o a s imple m a t t e r t o c a l c u l a t e t h e number c(G) of components of G from t h e subgraphs G-v .

I t is w e l l known and has been proved i n a v a r i e t y of ways t h a t every disconnected graph (of o r d e r a t least t h r e e ) is r e c o n s t r u c t a b l e . For any class of r e c o n s t r u c t a b l e graphs , t h e problem always remains t o f i n d t h e most e f f i c i e n t procedure t o a c t u a l l y de te rmine t h e graph G from i ts subgraphs G-v . I f G c o n t a i n s i s o l a t e d v e r t i c e s , t h e r e is l i t t l e d i f f i c u l t y i n de te rmining G . It is easy t o l o c a t e a subgraph G - w f o r which w is an i s o l a t e d v e r t e x of G . To produce G then , one need only add an i s o l a t e d v e r t e x t o G - w . Therefore t o develop an e f f i c i e n t procedure f o r de te rmining a d isconnec ted graph G from i t s proper maximal induced subgraphs , one can , w i thou t l o s s of g e n e r a l i t y , assume G t o be wi thou t i s o l a t e d v e r t i c e s . I n t h i s c a s e , t h e de te rmi- n a t i o n of "mul t ip le" components may be handled as fo l lows .

Theorem. Let G b e a d i sconnec ted graph wi thou t i s o l a t e d v e r t i c e s having k ) 2 components. Then G has e x a c t l y n components i somorphic wi th H , 2 5 n 5 k , hold: ( i ) f o r each v , G - v has a t l e a s t n - 1 components i somorphic wi th H , ( i i ) t h e r e exist v e r t i c e s v f o r which G-v has e x a c t l y n -1 components i somorphic w i t h H , ( i i i ) i f G - v has e x a c t l y n - 1 com- ponents isomorphic wi th H , t hen G - v has a component of o r d e r less than t h a t of H .

i f and only i f t h e fo l lowing t h r e e c o n d i t i o n s

We now o u t l i n e a procedure f o r de te rmining a d isconnec ted graph G from i t s subgraphs G - v . I t is based on one given by Harary [l]. AS w e have a l r eady no ted , G c a n , b e qu ick ly determined i f i t has i s o l a t e d v e r t i c e s . Thus w e hence fo r th assume G has no i s o l a t e d v e r t i c e s . The number of components of G i s given by

c(G) = min { c ( G - v ) l v E V(G)} = k .

Each subgraph G - v having k components is n e c e s s a r i l y ob ta ined by removing a v e r t e x v of G which is n o t a cu t -ve r t ex of G . We now cons ide r only these subgraphs .

Among a l l subgraphs G - v w i th k components, l e t G - u b e one having a component F of minimum orde r . Let F 2 , F 3 , ' * . ,Fk b e t h e

remaining components of G - u . N e c e s s a r i l y , each of t h e components

Fi, component of G . L e t w be a v e r t e x of F 2 , s a y , which is n o t a cu t -

v e r t e x o f F2 (of which t h e r e are a t least two). Then i n G - w (and

2 5 i 5 k , has o r d e r exceeding t h a t of F and each is a l s o a

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86 Annals New York Academy of Sciences

poss ib ly o t h e r subgraphs G - v as w e l l ) , k- 1 of t h e k components a r e

F - w , F 3 , * - * ,Fk. Consider t h e subgraphs G - v having t h e s e a s k - 1

of t h e i r k components. I f a l l such subgraphs have t h e same remaining

component, t hen t h i s component i s t h e miss ing component o f G , s a y F1.

I f , on t h e o t h e r hand, t h e r e are two o r more non-isomorphic subgraphs

G - v having t h e k - 1 components F - w , F3 , " ' ,Fk , t hen one of

t h e s e subgraphs has F - w as the remaining component and F = F - w

i s t h e miss ing component of G .

2

2

2 1 2

REFERENCE

1. F. Harary, Graphica l r e c o n s t r u c t i o n , A Seminar i n Graph Theory (F. Harary , e d . ) , Ho l t , R i n e h a r t , and Winston, New York (1967), pp. 18-20.