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### On the ascending subgraph decomposition problem for bipartite graphs

J.M. Aroca, A. Lladó, S. Slamin
2014 Electronic Notes in Discrete Mathematics
with 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 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

Hung-Lin Fu, Wei-Hsin Hu
2001 Discrete Mathematics
In this note, we prove that 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

Hung-Lin Fu, Wei-Hsin Hu
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 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 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

Carl Bürger, Jan Kurkofka
2021 Journal of Graph Theory
In a series 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]

Pierre Charbit, Fabien de Montgolfier, Mathieu Raffinot
2010 arXiv   pre-print
Using this tool we revisit 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

Pierre Charbit, Fabien de Montgolfier, Mathieu Raffinot
2012 SIAM Journal on Discrete Mathematics
Using this tool we revisit 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

Kai-Uwe Bux, Peter Smillie, Karen Vogtmann
2018 Journal of the London Mathematical Society
We give a simple construction 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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