AbstractLet (La,⊆) be the lattice of hereditary and additive properties of graphs. A reducible property R∈La is called minimal reducible bound for a property P∈La if in the interval (P,R) of the lattice La, there are only irreducible properties. We prove that the set B(Dk)={Dp∘Dq:k=p+q+1} is the covering set of minimal reducible bounds for the class Dk of all k-degenerate graphs

Mihók, Peter

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Discrete Mathematics 236 (2001) 273–279
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Minimal reducible bounds for the class of
k-degenerate graphs
Peter Mih%oka;b; ∗
aMathematical Institute, Slovak Academy of Sciences, Gresakova 6, 040 01 Kosice, Slovakia
bDepartment of Applied Mathematics, Faculty of Economics, Technical University, 04200 Kosice,
Letna 9, Slovakia
Received 30 October 1997; revised 26 November 1998; accepted 5 March 2000
Abstract
Let (La;⊆) be the lattice of hereditary and additive properties of graphs. A reducible property
R∈ La is called minimal reducible bound for a property P∈ La if in the interval (P;R) of the
lattice La, there are only irreducible properties. We prove that the set B(Dk) = {Dp◦Dq: k =p+
q+1} is the covering set of minimal reducible bounds for the class Dk of all k-degenerate graphs.
c© 2001 Elsevier Science B.V. All rights reserved.
MSC: 05C15; O5C75
Keywords: k-degenerate graph; Property of graphs; Additive; Hereditary; Vertex partition;
Minimal reducible bounds
1. Minimal reducible bounds
In this paper, we consider =nite undirected graphs without loops and multiple edges.
For unde=ned concepts, we refer the reader to [1,5]. A proper nonempty isomorphism-
closed subclass P of the class of all =nite simple graphs I is called additive and
hereditary if it is closed taking disjoint union and subgraphs of graphs in P, respec-
tively. Let us denote by (La;⊆) the lattice of all additive hereditary properties of graphs
ordered by inclusion (for details see [1,4]). Let P1;P2; : : : ;Pn be arbitrary hereditary
properties of graphs. A vertex (P1;P2; : : : ;Pn)-partition of a graph G is a partition
{V1; V2; : : : ; Vn} of V (G) such that for each i= 1; 2; : : : ; n, the induced subgraph G[Vi]
has the property Pi (for convenience, the empty set ∅ will be regarded as the set
inducing the subgraph with any property P). A property R=P1 ◦ P2 ◦ · · · ◦ Pn is
∗ Correspondence address: Department of Geometry and Algebra, Jessena 5, 041-54, Kosice, Slovak Republic.
Tel.: +42-95-622-1926.
E-mail address: mihok@kosice.upjs.sk (P. Mih%ok).
0012-365X/01/$ - see front matter c© 2001 Elsevier Science B.V. All rights reserved.
PII: S0012 -365X(00)00447 -7
274 P. Mihok /Discrete Mathematics 236 (2001) 273–279
de=ned as the set of all graphs having a vertex (P1;P2; : : : ;Pn)-partition. It is easy to
see that if P1;P2; : : : ;Pn are additive and hereditary, then R=P1 ◦P2 ◦ · · · ◦Pn is
additive and hereditary, too. An additive hereditary property R is said to be reducible
in La if there exist nontrivial additive hereditary properties P;Q such that R=P ◦ Q
and irreducible in La, otherwise.
For a given irreducible property P, a reducible property R is called minimal re-
ducible bound for P if P⊆R and there is no reducible property R′⊂R satisfying
P⊆R′. In other words, R is a minimal reducible bound for P in La if in the interval
(P;R) of the lattice La, there are only irreducible properties.
At the International Conference on Combinatorics held in Keszthely, Hungary, 1993,
dedicated to Paul Erdo˝s on his 80th, P. Mih%ok and B. Toft asked the following questions
(see [6, Problem 17:9]):
1. Is the factorization of every hereditary property into irreducible properties unique?
2. What are the minimal reducible bounds for the class of planar graphs?
The answer to question (1) has been found by Mih%ok, SemaniNsin and Vasky.
Theorem 1 (Mih%ok et al. [11]). Let R be an additive hereditary property of graphs.
Then the factorization of R into irreducible factors is unique apart from the order
of factors.
Theorem 1 provides an aOrmative answer to the problem of unique factorization
providing that the reducible property is additive. For non-additive reducible properties,
the answer, in general, is negative (see the example given in [11]).
Some partial results to problem (2) have been presented in [1–3,10]. However, the
question for the class of all planar graphs, seems to be very diOcult. In this paper, we
consider the analogous problem for the class Dk of all k-degenerate graphs.
A graph G is said to be k-degenerate if every subgraph of G has minimum degree
at most k (see e.g. [4,7,12,13]).
For k = 0, the class D0 is equal to the smallest element of La: O — the class of
all totally disconnected (edgeless) graphs. D1 is the class of all forests. The basic
properties of k-degenerate graphs have been introduced in [7]. Let us recall those,
which we shall use in the sequel.
Proposition 1. A graph G is k-degenerate if and only if the vertex set V (G) can be
labeled v1; v2; : : : ; vp such that in G[{vi; vi+1; : : : ; vp}]; deg vi6k for each i= 1; 2; : : : ; p.
A k-degenerate graph G is maximal k-degenerate if for every edge e of the com-
plement of G; G + e is not k-degenerate.
Proposition 2. Let G be a maximal k-degenerate graph of order p; p¿k + 1. Then
the minimum degree (G) of G is equal to k.
P. Mihok /Discrete Mathematics 236 (2001) 273–279 275
Proposition 3. Let G = (V (G); E(G)) be a graph of order p;p¿k + 1 and v∈V be
a vertex of degree k. Then G is a maximal k-degenerate graph if and only if G − v
is maximal k-degenerate.
Let us remark, according to the above presented propositions, that every maximal
k-degenerate graph can be obtained from the complete graph Kk+1 by a sequence of
adding of new vertices of degree k. We shall use the above presented structural results
to =nd the set of all minimal reducible bounds for Dk .
Let us recall that a property P∈ La is said to be degenerate if there exists a bipartite
graph not belonging to P (see [1]). Obviously, the class of k-degenerate graphs is also
degenerate since the complete bipartite graph Kk+1; k+1 
∈ Dk . It is proved in [9] that
every degenerate property is irreducible.
Let us denote by B(P) the set of all minimal reducible bounds for a property P∈ La.
It is worth pointing out that B(P) might be empty, in general. Also, if the set B(P)
is nonempty, there should be reducible properties R∗⊃P such that no R∈B(P)
“covers” R∗, i.e., there is no R∈B(P) such that R⊆R∗. We would like to prove
that both cases mentioned cannot occur, however these problems are still open. For
this reason, we say that the set B(P) is a covering set of minimal reducible bounds
for P∈ La if for every reducible property R∗;P⊂R∗, there exists an R∈B(P) such
that R⊆R∗.
Our main result is summarized in the following theorem.
Theorem 2. The set B(Dk) = {Dp ◦ Dq: k =p + q + 1} is the covering set of all
minimal reducible bounds for the class Dk of k-degenerate graphs.
Obviously, in order to prove that B(P) = {Ri: i∈ I} is a covering set of minimal
reducible bounds for P, it is suOcient
(1) to verify that Ri is a reducible bound for P; i∈ I ,
(2) to verify that there is no reducible property in the interval (P;Ri) for each i∈ I ,
(3) to prove that, if P⊆R for some reducible property R, then there exists an i∈ I
such that Ri⊆R.
An eSort to verify the steps may be met with resistance of diSerent kinds (see [2,8]).
In order to get over these diOculties, more information on the structure of reducible
properties in La is necessary. In the recent survey of hereditary properties [1], some
useful methods and results are presented. A new approach to realize step (2) for
k-degenerate graphs will be given in Section 2. A proof of step (3), completing the
proof of our main result, may be found in Section 3.
2. A Ramsey-type property of vertex-partitions
In [2], it was proved that the reducible property O◦D1 is the only minimal reducible
bound for the class D2 of all 2-degenerate graphs. The proof is based on the following
276 P. Mihok /Discrete Mathematics 236 (2001) 273–279
Fig. 1. The tree Tk .
constructions found by Broere: We recall them to motivate the idea of the proof of
our basic theorem.
Construction 1. For given graphs G and H , and the integer k, let G(H; k) denote the
graph obtained from G by joining all the vertices of k disjoint copies of H to each
vertex of G. Thus the graph G(H; k) is of order |V (G)|(1 + k|V (H)|).
Construction 2. The tree Tk (a complete k-ary tree) consists, as indicated in Fig. 1,
of ki−1 vertices on level i; i= 1; : : : ; k, and edges as indicated.
Construct a sequence of graphs G1; G2; : : : by
G1 =Tk and Gi =Gi−1(Tk ; k) for i= 2; 3; : : : :
For the graphs Gk constructed above the following lemma follows:
Lemma 1 (Borowiecki et al. [2]). (a) Gk ∈D2.
(b) For every partition V1; V2 of V (Gk); one has that Tk is a subgraph of Gk [V1]
or it is a subgraph of Gk [V2].
Since for every forest F there is a tree T such that F is a subgraph of T and for
every tree T there is an integer k such that T ⊆Tk , by Lemma 1, we have the following
corollary.
Corollary 1. For any graph F ∈D1; there is a graph G ∈D2 such that for every
partition {V1; V2} of V (G) there is a subgraph isomorphic to F in G[V1] or in G[V2].
In the sequel, if a graph G contains a subgraph isomorphic to a graph F then we
say that G contains F . Let Q; P1; P2 be any properties. We write
Q→ (P1;P2)
if for each pair F1 ∈P1 and F2 ∈P2 there exists a graph G ∈Q such that for every
partition {V1; V2} of V (G); G[V1] contains F1 or G[V2] contains F2 and we denote
this fact by G → (F1; F2). Hence the Corollary 1 yields D2 → (D1;D1).
The aim of this section is to prove:
P. Mihok /Discrete Mathematics 236 (2001) 273–279 277
Theorem 3. Let p; q be nonnegative integers. Then Dp+q → (Dp;Dq).
Proof. Without loss of generality, we can assume 06p6q. If p= 0, then Dq →
(O;Dq), because in general it holds.
For every Q∈ La; Q→ (O;Q). Indeed for any totally disconnected graph Dr of order
r and any graph F ∈Q we can take G to be the disjoint union of r copies of the graph
F to obtain G → (Dr; F).
To show, that Dp+q → (Dp;Dq) p¿1, it is suOcient to prove that there exists
a graph G ∈Dp+q; G → (F1; F2) for any pair F1 and F2 of maximal p-degenerate
and maximal q-degenerate graphs, respectively. We shall construct such a graph G by
induction using the following construction:
Construction 3. For given disjoint graphs G; F and S ⊂V (G) of cardinality |S|= r,
let J (G; F; S) denote the graph obtained from G by joining all the vertices of F to
each vertex of S. Thus the graph J (G; F; S) is of order |V (G)|+ |V (F)|.
Let G be a p+ q-degenerate graph, F a p-degenerate graph and suppose S ⊂V (G)
is of cardinality q, then by Proposition 1, the graph J (G; F; S) is p+q-degenerate, too.
Now, let us construct the graph G; G → (F1; F2), using induction on n= |V (F1)|+
|V (F2)|:
1. If F1 =Kp+1, and F2 =Kq+1, then G =Kp+q+1.
2. Next, let us prove that for any maximal q-degenerate graph F2; |V (F2)|¿q + 1,
there exists a (p + q)-degenerate graph G such that G → (Kp+1; F2).
By induction hypothesis, there is a (p + q)-degenerate graph G∗ such that G∗ →
(Kp+1; F2 − v) for a vertex v∈V (F2) of degree deg v= q. Let A= {Si⊂V (G∗); |Si|=
q; i= 1; 2; : : : ; s} be the set of all q element subsets of V (G∗), let G0 =G∗ and
Gi+1 = J (Gi; Kp+1; Si+1); i= 0; 1; : : : ; s−1. We claim that the graph G =Gs → (Kp+1; F2):
let {V1; V2} be any partition of V (G). Then either G∗[V (G∗) ∩ V1] contains Kp+1 or
G∗[V (G∗) ∩ V2] contains F2 − v. In this case, consider a set Sk ∈A such that G∗[Sk ]
is isomorphic to F2[NF2 (v)]. Then Gk+1 ⊆G and hence G[V1] contains Kp+1 or G[V2]
contains F2.
Now, let G∗∗ → (F1 − w; F2) for given F1; F2; |V (F1)|¿p + 1 and degF1w=p.
Let B= {Ti⊂V (G∗∗); |Ti|=p; i= 1; 2; : : : ; t} be the set of all p element subsets of
V (G∗∗), let G0 =G∗∗ and Gi+1 = J (Gi; F2; Ti+1); i= 0; 1; : : : ; t − 1. As in the previous
case, for any partition {V1; V2} of G =Gt , there is set Tk ∈B such that either G[V2]
contains F2 or G∗∗[Tk ] is isomorphic to F1[NF1 (w)] and since Gk+1 = J (Gk; F2; Tk+1)
is a subgraph of G; G[V1] contains F1 or G[V2] contains F2.
3. Covering set of minimal reducible bounds for the class of k-degenerate graphs
In this section, we are going to complete the proof of Theorem 2. Let us verify the
steps (1)–(3) described above:
278 P. Mihok /Discrete Mathematics 236 (2001) 273–279
(1) Every (p+q+ 1)-degenerate graph G is (Dp;Dq)-partitionable which is easy to
prove by induction on the number of vertices of G using Proposition 1 (cf. [1,2,6]).
Hence Dp+q+1 ⊂Dp ◦Dq.
(2) To proceed with step (2) we will mainly use Theorem 3 and the following
results:
Theorem 4 (Borowiecki et al. [2]). Let R1 ⊆R2 be additive degenerate hereditary
properties and suppose that P1 ◦ P2 ⊆R1 ◦ R2 for P1;P2 ∈ La. Then P1 ⊆R1 and
P2 ⊆R2 or P1 ⊆R2 and P2 ⊆R1.
The next lemma was observed by Broere:
Lemma 2. Let P;P1;P2;P3 and P4 ∈ La are properties satisfying P⊂P1 ◦P2 and
P→ (P3;P4). If P3 * P1; then P4 ⊆P2.
The proof of lemma follows from the following facts: let F1 ∈P3−P1 and F2 ∈P4.
Let the graph G ∈P be such that G → (F1; F2). Since G ∈P1 ◦P2 and F1 
∈ P1, the
graph F2 has property P2, implying P4 ⊆P2.
Let p; q be any nonnegative integers and R=Dp ◦Dq; 06p6q and k =p+q+ 1.
We are going to prove that in the interval (Dk ;R) of the lattice La there are no
reducible properties. Let Dk ⊂P1 ◦ P2 ⊆Dp ◦ Dq. Since by Theorem 3, Dp+q+1 →
(Dp;Dq+1), we have by the Lemma 2 that P1 ⊂Dp implies P2 ⊇Dq+1, a contradiction.
Therefore P1 =Dp and similarly Dp+q+1 → (Dp+1;Dq) implies that P2 =Dq.
(3) Let R=P ◦ Q be any reducible bound for the class of all k-degenerate graphs
Dk , i.e. Dk ⊂P ◦ Q.
Let i be any integer 06i6k. Then by Theorem 3 Dk → (Di ;Dk−i). By the Lemma 2
Di⊆P or Dk−i⊆Q.
Now, let p= min{j: Dj * P}. Note that 
= 0, since O⊆P for every P∈ La. If
p6k, then Dp−1 ⊆P and Dk−p⊆Q. If p¿k, then Dk−1 ⊆P and D0 ⊆Q. In both
cases, the inclusions Dk ⊂Di ◦Dj ⊆P ◦ Q such that k = i + j + 1 hold.
Acknowledgements
The author is very grateful to Mietek Borowiecki and Izak Broere for fruitful dis-
cussions and their help in the world of additive and hereditary graph properties. The
research was partially supported by Slovak VEGA Grant 1=4377=97 and by National
CEEPUS OOce of Hungary.
References
[1] M. Borowiecki, I. Broere, M. Frick, P. Mih%ok, G. SemaniNsin, Survey of hereditary properties of graphs,
Discuss. Math. Graph Theory 17 (1997) 5–50.
P. Mihok /Discrete Mathematics 236 (2001) 273–279 279
[2] M. Borowiecki, I. Broere, P. Mih%ok, Minimal reducible bounds for planar graphs, manuscript, 1997.
[3] M. Borowiecki, M. Ha luszczak, M. Skowro%nska, Minimal reducible bounds for 1-non-outerplanar graphs,
Report No. im-1-96, Institute Math., Technical Univ. Zielona G%ora, 1996.
[4] M. Borowiecki, P. Mih%ok, Hereditary properties of graphs, in: V.R. Kulli (Ed.), Advances in Graph
Theory, Vishwa International Publication, Gulbarga, 1991, pp. 42–69.
[5] R.L. Graham, M. GrWotschel, L. Lov%asz, Handbook of Combinatorics, Elsevier Science B.V, Amsterdam,
1995.
[6] T.R. Jensen, B. Toft, Graph Colouring Problems, Wiley-Interscience Publications, New York, 1995.
[7] R. Lick, A.T. White, k-degenerate graphs, Can. J. Math. 22 (1970) 1082–1096; MR42#1715.
[8] M. Borowiecki, I. Broere, P. Mih%ok, Minimal reducible bounds for planar graphs, Discrete Math. 212
(2000) 19–27.
[9] P. Mih%ok, Additive hereditary properties and uniquely partitionable graphs, in: M. Borowiecki, Z.
Skupien (Eds.), Graphs, hypergraphs and matroids, Zielona, G%ora, 1985, pp. 49–58.
[10] P. Mih%ok, On the minimal reducible bound for outerplanar and planar graphs, Discrete Math. 150 (1996)
431–435.
[11] P. Mih%ok, G. SemaniNsin, R. Vasky, Hereditary additive properties are uniquely factorizable into
irreducible factors, J. Graph Theory 33 (2000) 44–53.
[12] J. Mitchem, Maximal k-degenerate graphs, Utilitas Math. 11 (1977) 101–106.
[13] J.M.S. Simo˜es-Pereira, A survey on k-degenerate graphs, Graph Theory Newsletter 5 (75/76) No. 6,
1–7; MR55#199.


Minimal reducible bounds for the class of k-degenerate graphs 

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Minimal reducible bounds for the class of k-degenerate graphs

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