On minimal prime extensions of a four-vertex graph in a prime graph

Brandstädt, Andreas; Hoíng, Chınh T.; Vanherpe, Jean-Marie

doi:10.1016/j.disc.2004.06.019

Cited by 8 publications

(7 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now, by (9), (10) and Claim 4, y cannot be adjacent to v 2 , v 3 and v 4 , hence v 1 , v 2 , v 3 , v 4 , x and y induce an xbull. Claim 5 follows.…”

Section: Claim 5 P Has No 2-vertexmentioning

confidence: 92%

See 1 more Smart Citation

Efficient robust algorithms for the Maximum Weight Stable Set Problem in chair-free graph classes

Brandstädt

Lê

Ridder

2004

Information Processing Letters

Self Cite

View full text Add to dashboard Cite

“…Now, by (9), (10) and Claim 4, y cannot be adjacent to v 2 , v 3 and v 4 , hence v 1 , v 2 , v 3 , v 4 , x and y induce an xbull. Claim 5 follows.…”

Section: Claim 5 P Has No 2-vertexmentioning

confidence: 92%

“…Lemma 1 describes all minimal prime extensions of the co-gem which is a useful lemma when considering the MWS problem in certain graph classes as we will see in the next section. Lemma 1 was given in [4] with a long proof using a rather complicated technique described in [27] and in [3] using a lemma from [26].…”

Section: The Co-gem Lemmamentioning

confidence: 99%

Efficient robust algorithms for the Maximum Weight Stable Set Problem in chair-free graph classes

Brandstädt

Lê

Ridder

2004

Information Processing Letters

Self Cite

View full text Add to dashboard Cite

“…However two different permutations (such as 4123 and 2341) can give rise to the same graph (a star graph on 4 vertices in this case, sometimes called a 3-claw) and so the connection is quite weak. In particular the finite type property is not preserved (for example, 4123 and 2341 both have infinite type whereas the 3-claw has finite type [3]). …”

Section: Proposition 11 a Pattern Class Is Substitution-closed If Amentioning

confidence: 93%

“…Similar notions have recently been studied in graph theory [3,[8][9][10][12][13][14][15]. For graphs there are natural analogues of the notions of 'basis', 'interval' and 'simple' and the substitution closure has been a tool in graph theory since Lovász's work [11] on perfect graphs.…”

Section: Proposition 11 a Pattern Class Is Substitution-closed If Amentioning

confidence: 93%

Substitution-closed pattern classes

Atkinson

Ruškuc

Smith

2011

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

show abstract

“…Find necessary and sufficient conditions for Z * to be finite. Various researchers investigated Problem 2 and many sufficient conditions have been presented [5,6,10,11,13,16,17]. These, and other similar papers, give forbidden subgraph characterizations of the closure under substitution of various classes of graphs.…”

Section: Motivationmentioning

confidence: 99%

All minimal prime extensions of hereditary classes of graphs

Giakoumakis¹,

Olariu

2007

Theoretical Computer Science

View full text Add to dashboard Cite

The substitution composition of two disjoint graphs G 1 and G 2 is obtained by first removing a vertex x from G 2 and then making every vertex in G 1 adjacent to all neighbours of x in G 2. Let F be a family of graphs defined by a set Z of forbidden configurations. Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathematics 177 (1997) 83-97] proved that F * , the closure under substitution of F, can be characterized by a set Z * of forbidden configurationsthe minimal prime extensions of Z. He also showed that Z * is not necessarily a finite set. Since substitution preserves many of the properties of the composed graphs, an important problem is the following: find necessary and sufficient conditions for the finiteness of Z *. Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathematics 177 (1997) 83-97] presented a sufficient condition for the finiteness of Z * and a simple method for enumerating all its elements. Since then, many other researchers have studied various classes of graphs for which the substitution closure can be characterized by a finite set of forbidden configurations. The main contribution of this paper is to completely solve the above problem by characterizing all classes of graphs having a finite number of minimal prime extensions. We then go on to point out a simple way for generating an infinite number of minimal prime extensions for all the other classes of F * .

show abstract

On minimal prime extensions of a four-vertex graph in a prime graph

Cited by 8 publications

References 18 publications

Efficient robust algorithms for the Maximum Weight Stable Set Problem in chair-free graph classes

Efficient robust algorithms for the Maximum Weight Stable Set Problem in chair-free graph classes

Substitution-closed pattern classes

All minimal prime extensions of hereditary classes of graphs

Contact Info

Product

Resources

About