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On the ascending subgraph decomposition problem for bipartite graphs

2014
*
Electronic Notes in Discrete Mathematics
*

with

doi:10.1016/j.endm.2014.08.004
fatcat:k7lh5akv45ejfirj2wlfg7egiu
*star**forests*. ... We show that every bipartite graph G with n+1 2 edges such that*the*degree sequence d 1 , . . . , d k*of*one*of**the*stable sets satisfies d i ≥ n − i + 2, 1 ≤ i < k, admits an*ascending**subgraph**decomposition*... By*the*column sum property*of**the*matrix C,*the**star**forest*F s has i c is = n − s + 1 edges and, by*the**ascending*column property, it is isomorphic to a*subgraph**of*F s−1 . ...##
###
On star-forest ascending subgraph decomposition
[article]

2015
*
arXiv
*
pre-print

,d_k

arXiv:1512.02161v1
fatcat:ds2ky6y6jbe5vlgyl5a2ndflgu
*of*one*of**the*stable sets satisfies d_k-i> n-i for each 0< i< k-1" admits an*ascending**subgraph**decomposition*with*star**forests*. ...*The**Ascending**Subgraph**Decomposition*(ASD) Conjecture asserts that every graph G with n+1 2 edges admits an edge*decomposition*G=H_1⊕...⊕ H_n such that H_i has i edges and it is isomorphic to a*subgraph*... Let C be*the*(k × t) matrix whose entry c ij is*the*number*of*edges incident to a i in*the**star**forest*F ′ j*of**the*edge*decomposition**of*G R . ...##
###
Page 1302 of Mathematical Reviews Vol. , Issue 91C
[page]

1991
*
Mathematical Reviews
*

A tree C is called a caterpillar if

*the*removal*of**the*end-vertices from C produces a path. A*star*-like graph is a tree obtained from a*star*by subdividing*some**of*its edges. ... A graph G with ty ) edges is said to have an*ascending**subgraph**decomposition*(ASD) if E(G) can be partitioned into n sets E,, E>,---,E, such that |E;| =i for 1 < i <n and*the**subgraph*induced by E; is ...##
###
A note on ascending subgraph decompositions of complete multipartite graphs

2001
*
Discrete Mathematics
*

In this note, we prove that

doi:10.1016/s0012-365x(00)00171-0
fatcat:zsp32tj4c5chvg5xungpbxjeqi
*the**ascending**subgraph**decomposition*conjecture is true for complete multipartite graphs. ... Acknowledgements*The*authors would like to express their appreciation to all*the*referees for their helpful comments. ... In this note, we shall prove that every complete multipartite graph does have an*ascending**subgraph**decomposition*. In order to prove*the*main result, we need two deÿnitions and several lemmas. ...##
###
Page 659 of Mathematical Reviews Vol. , Issue 91B
[page]

1991
*
Mathematical Reviews
*

*The*paper lists known results that support this conjecture and gives a proof

*of*a new such result: Every

*forest*with ("}') edges has an

*ascending*

*decomposition*into n

*star*

*forests*. ... J. (1-EMRY)

*Ascending*

*subgraph*

*decomposition*for

*forests*. Proceedings

*of*

*the*Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989). Congr. ...

##
###
Page 2778 of Mathematical Reviews Vol. , Issue 97E
[page]

1997
*
Mathematical Reviews
*

Harris (1-APLS; Boone, NC)
97e:05155 05C70
Chen, Huaitang; Ma, Kejie (PRC-QTC-OR; Qufu);
Zhou, Huishan (1-GAS; Atlanta, GA)

*The**ascending**star**subgraph**decomposition**of**some**star**forests*. ... Then G is said to have an*ascending**star**subgraph**decomposition*if G can be decomposed into n*subgraphs*G), G2,---,G, such that each G; is a*star**of*size i with 1 <i <n. ...##
###
Ascending subgraph decompositions of regular graphs

2002
*
Discrete Mathematics
*

We prove that every regular graph with ( n+1 2 ) + t edges, 0 6 t ¡ n + 1, can be decomposed into n

doi:10.1016/s0012-365x(01)00445-9
fatcat:mbjkmwxhevenhbxkyh7zhayrqq
*subgraphs*G1; G2; : : : ; Gn such that |E(Gi)| = i and Gi 6 Gi+1 for i = 1; 2; : : : ; n − 1 and |E(Gn ... Acknowledgements We thank*the*referees for their helpful comments. ... One*of**the*reasons that this*decomposition*problem is interesting can be seen from*the**decomposition**of*a*star**forest*into*stars*. ...##
###
Page 6439 of Mathematical Reviews Vol. , Issue 95k
[page]

1995
*
Mathematical Reviews
*

*The*authors prove that a

*star*

*forest*

*of*size ("5 ') possesses an

*ascending*

*star*

*subgraph*

*decomposition*if

*the*size

*of*each component is at least n. R. H. ... Then G is said to have an

*ascending*

*star*

*subgraph*

*decomposition*if G can be decomposed into n

*subgraphs*G), G2,---,G, such that each G; is a

*star*

*of*size i with 1 <i <n. ...

##
###
Page 4369 of Mathematical Reviews Vol. , Issue 90H
[page]

1990
*
Mathematical Reviews
*

*Some*recent results have established

*the*validity

*of*

*the*conjecture for

*some*special classes

*of*graphs with ("3') edges:

*star*

*forests*, graphs with at most n +2 vertices, and graphs

*of*maximum degree d( ... and

*the*

*subgraph*G;

*of*G induced by E; is isomorphic to

*some*

*subgraph*

*of*

*the*graph G;,, induced by Ej,;, i= 1,---,n- 1. ...

##
###
Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions
[article]

2017
*
arXiv
*
pre-print

In

arXiv:1508.04718v2
fatcat:mztzhrisyzb4pew7nhtkerrhrm
*the*companion paper [Linear rank-width*of*distance-hereditary graphs I. ... Also, we give a simpler way to obtain*the*known vertex-minor obstructions for graphs*of*linear rank-width at most 1. ... Acknowledgment*The*authors would like to thank Isolde Adler for initial discussions on this problem. ...##
###
Duality theorems for stars and combs III: Undominated combs

2021
*
Journal of Graph Theory
*

In a series

doi:10.1002/jgt.22781
fatcat:eahkrlf6evghppq5fu7z7ewzxm
*of*four papers we determine structures whose existence is dual, in*the*sense*of*complementary, to*the*existence*of**stars*or combs. ... Here, in*the*third paper*of**the*series, we present duality theorems for a combination*of**stars*and combs: undominated combs. ... ACKNOWLEDGEMENT We are grateful to*the*reviewer for helpful comments and an efficient review. Open Access funding enabled and organized by Projekt DEAL. ...##
###
Page 6823 of Mathematical Reviews Vol. , Issue 2000j
[page]

2000
*
Mathematical Reviews
*

We say G has an

*ascending**subgraph**decomposition*(ASD) if*the*edge set*of*G can be partitioned into n sets generating graphs G), G2,---, G, such that |E(G;)| =i (for i =1,2,---,m) and G; is isomorphic ... Two extreme cases*of*simply sequential numberings (universal simply sequential numberings and stiff labelings) are considered for*stars*K),,*the*galaxy K,,UK),,*the*galaxy qP2,*the**forest*P;UP,, and*the*...##
###
Linear Time Split Decomposition Revisited
[article]

2010
*
arXiv
*
pre-print

Using this tool we revisit

arXiv:0902.1700v3
fatcat:mp2o4g7isjdldk7nplyzi5pxda
*the*problem*of*designing a simple linear time algorithm for undirected graph split (also known as 1-join)*decomposition*. ... Given a family F*of*subsets*of*a ground set V, its orthogonal is defined to be*the*family*of*subsets that do not overlap any element*of*F. ... Acknowledgment We wish to thank Vincent Limouzy for his participation to*the*earlier studies we did on*the*subject. ...##
###
Linear Time Split Decomposition Revisited

2012
*
SIAM Journal on Discrete Mathematics
*

Using this tool we revisit

doi:10.1137/10080052x
fatcat:vokwf42zu5donnfklcfbajj63q
*the*problem*of*designing a simple linear time algorithm for undirected graph split (also known as 1-join)*decomposition*. ... Given a family F*of*subsets*of*a ground set V , its orthogonal is defined to be*the*family*of*subsets that do not overlap any element*of*F . ... We wish to thank Vincent Limouzy for his participation in*the*earlier studies we carried out on this subject. ...##
###
On the bordification of Outer space

2018
*
Journal of the London Mathematical Society
*

We give a simple construction

doi:10.1112/jlms.12124
fatcat:kuc7a5glajcltjhjfj7edefnma
*of*an equivariant deformation retract*of*Outer space which is homeomorphic to*the*Bestvina-Feighn bordification. ... This results in a much easier proof that*the*bordification is (2n-5)-connected at infinity, and hence that Out(F_n) is a virtual duality group. ... Note that every*subgraph**of*G contains a unique maximal core*subgraph*. If this maximal core is empty*the**subgraph*is a union*of*trees, i.e. a*forest*in G. ...
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