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Wenzel, Günter H.

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On Gallai's decomposition theorem for graphs

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1
Nr.65. 1986
On Gallai', Decompo,ition Theorem (or Graph,
von
Günter H. Wenzel
(Universität Mannheim)
_: •••••.••• h •• _._._ •
•
i:,
-'~ •..
A note on Gallai's decomposition theorem for graphs
Günter H. Wenzel
(Mannheim)
91. Introduction
T. Gallai (3) proved a decomposition theorem for finite graphs in
terms of their quasimaximal strongly autonomous vertex sets. It was
reviewed in D. Kelly's review paper on "Comparability graphs". (5).
Apart from reviewing existing results on comparability graphs
D. Kelly generalized these results, as far as possible, to the case
of infinite graphs. Gallai's theorem is not among them. The theorem is,
however, true fora large class of infinite graphs - the so-called
"Non Limit - graphs", shortly "NL-graphs". Although partial aspects
of the resült can be found in W.H. Cunningham & J. Edmonds (2) and
L.N. Shevrin & N.D. Filippov (6) (both papers make no reference to
Gallai's original paper) there is no theorem in the literature that
displays all features of Gallai's useful theorem in the infinite
case:as, e.g. the role of strongly autonomous subsets and their full
interaction with Gallai's "edge-classes". Therefore'we formulate it
(Theorem 1) and give a short direct proof. It is based bc:)t::hon the" edge-
vertex-lemma" (Lemma 1) that Gallai derived (I quote) as a"remarkable
consequence" from his finitary decomposition theorem and on a useful
lemma of D. Kelly (Lemma 2). In this way a new proof arises even for
the. finitary case. As indicated above, parts of theorem 1 can be. found
in L.N. Sh~vriri & N.D:~Filippov (6) ( see 93, Proposition 3 and
94, Lemma 16 ) in a different language. They use Zorn's Lemma, we don't.
Theorem 2 appears as a natural and.new supplement of Gallai's' theorem:.
In order to formulate it, we define a "Gallai-decomposition" as a
maximal dec6mposition into autonomous subgraphs such that the "external
edge-classes" constitute a single edge-class. Theorem 2 states that
exactly the NL~graphs have such a decomposition.
Two examples illustrate how the infinite version of Gallai's theorem
(i.e. theorems 1 and 2) can be usefully applied. The note ends with
-1-
aremark on graphs G with connected G and G that da not contain P4'
This remark analyses an example of D. Kelly in (5). (The referee
pointed out to me that R. M6hring announced ~ related result in
Methods of Operations Research 45 (1983), 287-291.)
As for the concepts "autonomaus", "strongly autonomo~s", "in general
pos ition", "edge-class" etc... see D. Ke11y (5).
If G is a graph, V(~) denot~s the'set of vertices of G and E(G) the-set of edges ofG. G denotes the complemented graph of G.
"NL-graphs" are graphs thatcontain at least one quasimaximal
strongly autonomaus set-Q-'of vertices (i.e. Q is maximal among all
proper strongly autonomaus subsets of V (G)). For A So V (G), G [A)
denotes the full subgraph of G with vertex set A. If T is a partition
of V(G) into autonomaus subsets then G~ denotes the corresponding
quotient graph. For a,b~ V(G),"a,.",b"indicates the existence of the
edge ab. For A,B~V(G),"A""B" means "a-b for all a EA and bE'B" and
"A + B" means "a.,Lb for all a ~A and b~ B".
'2. The decomposition theoremand ap~lications
~eorem 1 (Gallai's decomposition theorem):
Let G be anarbitrary non-trivial NL-graph:
iIl V(G) is the disjoint union of all its quasimaximal strongly
autonomaus subsets A., i EI.
l
(1LJ..Ql := fAi1i.~I~ is called the "canonical decomposition of
G". An edge class D is called "internaI" if D£ E (G[Ai]) for
same i, otherwise "external".)
iII) <r/( G) is 0btained as fallows:
ill If G is not connected, then lI(G) consists of the com =
ponents of G. There is no external edge class.
ill If TI is not connected, then 'lI(G)consists of the com=
ponents of G. For any fixed i,jeI, i;ij, we have A.r"'JA.,
l J
and the set E.. of all-lJ-
edge class.The classes
A.A. -edges con~t~tutes an
l J
E .. are exac~lyall externallJ
edge classes.
121 If G and Gare connected, then ~(G) is the unique largest
partition of V(G) into proper subsets which satisfies
the following two properties (a), (b):
ill A.r-.JA.orA.r-/JA. for all i,j e-I, i;ij.
l J l J
l.!U-~he set C:=V(E ..;i,je-I,i;ij) is a single edge class
. \ _ lJ
--~fn _t hi s .c~se. Y\CJ ..,-:=; ',VJ.~J)~-1~~~.7.~;;~~2';'=i::"_.:._
••. ~"1.",;"".;~"::::::::,"'=.,,,,_:_,,,,,',;3 .•.:.r_.,,_\::.: ..•~~., .•i"'''';''--'"'',~,~,;;.•.~~J::t""rt".;'!-:;~'2'.~?:-'~.r.:;j;"--"~':iLJ<:!~;;:~~-:.~[."'-~r.Mi!p "iF'»'?'"r ...•:r".~*.i<!.£11'~, ,_~~"'l"~~"'-Y'!':~'Y!':-o'~
It is possible to give a short proof of the next lemma based on
the "forbidden pattern lemma" (Arditti & Jung (1) and
Gilmore& Hoffman (3), corollary 1 to Lemma 3). We leave it as an
exercise.
Lemma 1 (Edge-Vertex-Lemma of Gallai) : If G is a graph and E1,E2
We include a proof of the next lemma, since the original proof in
D. Kelly (5) contains a minor mistake.
Lemma 2( D. Kelly (5)): Let A £ V (G) be a non-trivial, independent
and autonomous vertex set of the graph G. There exists a strongly
autonomous vertex set B wi th (i) A s: B, (ii) (B'\. A) 7:-' A.
proofof lemma 2:
If A is strongly autonomous, we take 13 = A. Otherwise let
F' = [Ci; i f- I] be the set of all autonomous subsets of V(G) that
are in general position with A and form H:=U(Ci; ie I).
B\: = HvA is autonomous and A;: 13. We have (13\ A) ~A, since
(C.\ A)rfA is eas'ily seen to hold true for every it I. We will
l
show that B is strongly autonomous and are done:
Assurne that there is some autonomous X in general position with
]ß;. If xn A = VJ, then XnH" VJ, i.e. X()Ci 1= VJ for some i ~I.
Thus, XlJCi = Cj f 13for some j £: I, a contradiction.
We conclude X ~ A (otherwise X~F, i.e. X.£ B which is impossible).
(B '\A) l' A implies (X" A) ,;-.A from which we deduce that L. :=C.v(X\A)
l l
is aut;onomous. LiEF implies Li ~ Bi hence :x ~ B, a contradiction.
q.e.d.
Eroof of theorem 1:
iIl folIows, as in the finite case, immediately from the fact
that two strongly autonomous subsets X,Y£V(G) are either diSjoint
or comparable.
-3-
i
I
mous subsets,lI(G) consists of the components of G. Hence,
A.rJA. for all i,j~I, ifj. This fact shows that abAbc, aE:Ai,1 J
bE-A. (ifj) implies CE A., i.e. [abJ::..s.E...• If cEA. is arbitrary
J 1.. lJ 1
with cfa, then a and c are in the same component of G~G(b)]
(where NG(b) denotes the neighborhood of b in G), hence ab Abc.
Thus, E.. =(abl:;. ffi: We first show that'i/(G) has properti.eslJ
(a) and (b). (a) is clear. As to (b): If abEErs' then abAbc
implies crAs' hence {ab)==S.C. Since V(labJ~) = V(Ai;icI1) for
s ome I1 £ I wi tb:., {r, s5 c. I1, we' swi tch over to G'TIvia the pro j ection
p: G ---ol) G~ and study 1:== {Ai; i E-I1\. We are done if we can prove
that T ~ll(G) is impossible. For then V([abJ=.) = V( [cd7=) = V(G)
for all cdE-Euv (u,vE-I, UfV), and lemnm 1 settles the matter.
Thus, assume T?~G). SincelTl~2, T is not strongly auto=
nom6us (otherwise p-1 (T) would be strongly autonomous in G).
Let 1be an autonomous subset of rr(G) in general position with T
and consider X(\T = {Ai;iE: I2~ with.I2£.I1• X(\Tand T\X are auto=
nomous. Thus ,nei ther X() ~ nor T,\ X contain.an ,e.dg,~from tab]::,;
am X"T.-(T,X), i.e. Xf\T".(T,X) in 'G1ifWecanapply case' (2)
of our theorem to the graph J![T'j ," since its complemented graph
i s not connected. The edge classes of ci1i[TJfall into two ca te=
gories: internaIones and external ones.[abJ= cannot be ani.nternal
one, since V([ab]:=.) =VT. Thus, it is external and -(again since
V([abJ::) =VT)- the only external one. Thus, XI1T andT' X are the
two components of .Gii[T];and [ab]==.c.onsists exactly of, all the
(X!)T,T'\.X)-edges. Lp., every autonomous set of Gi'""intersecting
both Xf)T and T\ X must c<Dltainall of T. Thus, there is no
a utonomous Y in general position with X()T, resp. T" X, since
otherwise YtJ(T'\X), resp.YU (XnT) would be autonomous.
I
i
We conclude that XI"\T and T'\ X are strongly autonomous, i.e. they
are singleta. Thus, T consists of 2 elements. Hence, T isinde=
pendent in ~andautonomous. By lemma 2, there exists a strongly
autonomous set D with T,£D and T+(D'\.T) in G1l"'.SinceGII is connec~1
I
ted, we get D~fjJ'"(G),i.e. ~TI = ID(' = 1, a contradiation.
T'hus, T ='if( G), and we are done., -4-
~~~:rri1[~Y~~~~1!Er:l~~~~~~o~~~~:~~.L~;~=~*.t~:t!f~~6~~~-if.:'87~~;tL~~,~~.g~~~i)1f~:~}!~~.~:~~~;{-}~~~tt~~~~:?l;Ii?~:~..;~.;;:~,:W~~~~~T;f~;;;:;;~ ':?:~$r- 'Fl=::'~._M;fo ;~~~ T~ '' 7~~~'.~/.~~ Y'':'~
In order to prove the fact that e-tr (G) is thefhhiquel larg.e>~t
partition with properties (a),(b), we choose same partition
f (G) = {Bj;jf. J) of V.(G) satisfying (a),.(b) anddenoteby g
the set of all BiBj -edges. No Bj is in general position with
any Ai' Hence, every Bi is either the disjoint union of suitable
A.'s or it is contained in same Ak which then, in turn, is theJ
disjoint union of sui table BI' s. Rn c I: VJ. is clear, hence R=C.
Assume JB:j = V(Ai; i E- 11) for same 11 £ I wi th '11 { ?- 2. Then
{ 1 11C = R forces the independence of Ai; i E:- 11 J in G . By lemma 2,
we obtain a strongly autonomaus D £7f(G) with {Ai;i€- 11\£ D.
The connectedness of G~forces D I:T(G) contradicting the fact
that Gl1has no non-trivial strongly autonomaus vertex sets.
H~ence, f (G) tf T (G)• g • e.d.
Let us call a decomposi tion ~ (G) '={ Ai; i E I1 of a graph G a
"Gallai-decomposition" if it is maximal among all partitions
satisfying the following two properties:
~ A.,-,JA.or A.rf'A. for all i,j E-1, il:j.
1 J 1 J
l£lThe set of all AiAj-edges (i,j<;1, il:j) constitutes a
single edge classQ. Ci.p. II \~2).
'Rhe next theorem constitutes a natural supplement of Gallai's
decomposition theorem.
Theorem 2: Let G be a graph. and f (G) a Gallai-decomposi tion.
Then G is an NL-graph and we have the following situation:
ill V(C) is a component of G and V(C) = V{AjJi e 11) for same
11S:1. 1f 11;' I, then .f (G) = lAi;i€: 1-t)V{D) where D is the
jain of all components I: V(C).
ill 1f G i8 connected, then f(G) =1i'(G), and all proper
autonomaus suhsets of V(G) are contained in a quasimaximal
strongly autonomaus set.
-5-
proof:
(1 ):Assume that G is not connected and Cj' j~ J, are the com=
ponents. Clearly, V(C) =V(Ai;i€IAAi(")V(C) 1= ~). Since V(C)
is connected, there is some jOt::-J wi th V(C) ~ C. and, hence,
JO
V(C) = C.• Since 6'"(G):= {A.; A.~ C .)v fU(C .;je-J, jl=j01 isJO l l JO L J )
. a partition satisfying (a),(b) and since .f(G)~G(G)., we con=
clude f(G) = 6'"(G).
(2): If G is corinected, we have V(C) = V(G) as the only compo=
nent. Let D ~~_~?_~!bi~~a~y autonomous se~that intersects both
A. and Al'1 properly'fOr some_~O l i1' • Then A. VA.' islO lO l1 ..
A.lO
properly. Therefore we have either D = LJ(Ai;iE 11) for some
I1~I or D = (DnAio)v(V(Ai;'i~I2) for some iofI2<; I with
Cf 1= D("\A.. ~ A.• In neither case can D contain an A.A.-edge,
. lO lO l J
if D is a proper autonomous subset. For such D we have either
autonomous and, hence, Ai ~A. (otherwise V(G) = V(C)s..D) .
. -0 l1 _
Hence, G(G) := {AiOUAi1~ v fAi;i€ I'fiO,i1\J is a partition of
V(G) satisfying (a), (b) and 6' (G) >f(G), a contradiction.
Thus, every autonomous set DSV(G) intersects at most one
iJll or in {Dv AiJu fAi; iOl=i.~ I" 12) a
partition of V(G) satisfying (a),(b). Hence, D = Ai ' resp.
o
Ai of .F (G)D~A. depending on the case. Therefore the setslO
are ( exactly all ) quasimaximal strongly autonomous subsets of
V(G), and each proper autonomous set D is contained in one of
them. g.e.d.
W€ contended in ~1 that the extension of GaIlai's theorem to
NL-graphs (theorem 1) and the complementing theorem 2 con=
stitute the natural basis of othsr results also in the infinite
case. To make out my case, let us prove two interesting results
of D. Kelly ([2 J) in the light of our theorems:
1} Lemma3.2 in [2J reads (rephrased). as folIows: "1f G is a
graph and A£ V(G) an autonomous, connected subset, then each
of the components ofG[A] is a strongly autonomous subset of
V (G ) • 11
..•6-
Qroof': Let t13i;i €; 11 be.the eomponents of A in G[A]. \'I.l.o.g.
\11)2 (otherwise let X and A be in general position and auto=
nomous in G; then A {'\X and A" (X "A) 'are autonomous in G,
a eontradietion). By theorem 1, the Bi are the quasimaximal
strongly autonomous sets of G CA], we have Bi,vBj for all i;fj
and, for eaeh ilj, the set F .. of all 13.13.-edges eonsti tutes a
1J 1 J -
single edgeelass of G (A] and, henee, of G. Fix iOcI and let
C be an autonomous vertex set of G properly interseeting B. .
10
Then C ()A = C nBio and {EiOUCJufBi; i6 I'[id]are the eomponentf
of G fÄ u cjJ. Thus, the (E. V C, B. )-edges and the 13. B.-edgesr 10 1~ 10 1
eoineide, sinee both sets eonstitute the same edge elass by
theorem 1. We eonelude c~ B. • g.e.d.
.. 10 ---
2) Theorem 3.4 in [2] reads as folIows: " A non-trivial eonneeted
graph G with a eonneeted eo~plement Ghas no proper autono=
mous vertex sets if it has no proper strongly'a.utonomous
,vertex sets."
Qroof: G is an NL-graph with trivial eanonieal deeomposition.
Theorem 2.2 proves the result. g.e.d.
~3. Aremark on P4~
Let Pn denote the graph with n vertiees 8.1,a2, ..... ,an and
edges a1a2, a2a3, , an_1an. D. Kelly [2J refers to
various proofs for the faet that finite prime graphs (i.e.
graphs without proper autonomous vertex sets) eontain P4 as
a full subgraph (we wri te P4 ~ G).
In the finite ease the property P4 ~ G holds even true for any
"
graph G such that G and Gare connected, and this is the basis
of the ;finitary prbofs. D. Kelly generalized the. resul t to
1
ib1
• a.,
diagram 2
for all m< n;
for all m< n;
b ",an n
an"i'am, bm
b rJa b
n m' m
for all n €\N.
o 0We denote by G , R , G, Rn the follow'ing graphs:n nn
Gn := G~IJ{a1,...,anJv fb1' ,bn~J;
Rn .- Gl-Jo[{a1,..•,an+1)lJ{b1, ,bn)];
G~ : = GNo[fa2, ..• ,an+1Jufb1 , •.•• ', bn+11J
R~ : = G
Ho
(fa2, ••• , an+1)vfb1 ' ••.. , bn 3J
Clearly', G- = RO and H = GO •n n n-n
more. Thus, an alltogether new proof was
asked fore Kelly's example (see [2] ) is
essentially the ordered set of diagram 3.
Its comparability graph G is anL-graph
such that G and Gare connected andP4]tG.
This comparability graph G is the graph
Q:.H
o
- on thevertex set f an;nt-INJvfbn;n~ \NI
defined as followe (see diagram 4):
infinite prime graphs and showed via an example that the
connectedness of G and G do not suffice any
Remark: Let G be a non-trivial graph such that G and TI are con=
nected:. P4 ~ G imp~ies GNot:: G.
Thus, Kelly' s exarr.pleis the minimal model for .his case.
Apart from this observation the result subsumes all finitary
proofs.
-8-
~roof of the remark:
Since P4 = P4 we have P41G. Gi course, P3~G since G andG
are connected. Let GGt,b,c1 = P3 with a"'" c-b be given. Then
a "'I'C, b~c in G, and we choose a shortest path (in G) from
c to b. Since P41G, it is of the form c""e-p with a",e in G.
G(a,b,c,eJ~ G shows G [a,c,e] = H1~ G.
Assume H (see diagram 4) ~ G. Since G is connected, we haven
a path an+1 ~ bn+1"" bn in G. Let gE fbj; j ~ nJv 1 aj; j ~ n J.
If bn+1~g, then G[an+1,bn+1,bn,g] = P4~ G, a contradiction.
Thus, bn+1~ g for all of the aboveg. Hence, G[{aj,bj; j~n+131
= Gn+1~ G. Then Gn+1= H~+1 ~ G . Since TI is connected, we have
a pa th of length 3, say bn+1 ""an+2 '" an+1, from bn+1 to an+1
in G. As above,G [{ai;i~on+2} ~Jbjo; j:~ ~~1J] =G~+1~ G.
-;;-C} 0- 00 0
Hence, G -1 = H".+01~ G. ßy our c onstruction, H o~.H +1~ G.n+ TI.o n n
Thus, G\..I = U(H ;n€:-\N) ~G. g.e.d.
(\0 n
References:
(1) J.C. Arditti & H.A. Jung (1980) : The dimension of finite
and infinite comparability graphs,
J. London Math. Soc. (2) 21, 31 -38
(2)W.H. Cunningh,am & J.Edmonds (1980): A combinatorial
decomposition theorY'i'
Can. J. Math., Vol. XXXII, No. 3, pp. 734-765
(3) T. Gallai (1967): Transitiv orientierbare Graphen,
Acta Math. Acad. Sei. Hungar. 18, 25-66
(4) P.C. Gilmore & A.J. Hoffman (1964): A characterization of
comparability graphs and of interval graphs,
Can. J. Math., Vol. XVI, pp. 539-548
(5) D. Kelly (1985) : Comparability graphs, in "Graphs and Order",
ed. I. Rival, Dordrecht-Boston, pp. 3-40
(6) L.N. Shevrin & N.D. Filippov: (1970): Partially ordered sets
and their comparability graphs,
Siber. Math. J. 11, pp. 497-509
-9-


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On Gallai's decomposition theorem for graphs

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