scholarly journals The Erdős–Pósa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces

2007 ◽  
Vol 307 (6) ◽  
pp. 764-768 ◽  
Author(s):  
Ken-Ichi Kawarabayashi ◽  
Atsuhiro Nakamoto
Keyword(s):  
Edge Disjoint ◽  
Odd Cycles ◽  
Orientable Surfaces

scholarly journals Edge-disjoint odd cycles in 4-edge-connected graphs

2016 ◽  
Vol 119 ◽  
pp. 12-27 ◽  
Author(s):  
Ken-ichi Kawarabayashi ◽  
Yusuke Kobayashi
Keyword(s):  
Connected Graphs ◽  
Edge Disjoint ◽  
Odd Cycles

scholarly journals Edge-disjoint odd cycles in planar graphs

2004 ◽  
Vol 90 (1) ◽  
pp. 107-120 ◽  
Author(s):  
Daniel Král’ ◽  
Heinz-Jürgen Voss
Keyword(s):  
Planar Graphs ◽  
Edge Disjoint ◽  
Odd Cycles

scholarly journals Dense Graphs With a Large Triangle Cover Have a Large Triangle Packing

2012 ◽  
Vol 21 (6) ◽  
pp. 952-962 ◽  
Author(s):  
RAPHAEL YUSTER
Keyword(s):  
Absolute Constant ◽  
The Other ◽  
Asymptotically Optimal ◽  
Edge Disjoint ◽  
Other Hand ◽  
Dense Graphs ◽  
Large Triangle ◽  
Odd Cycles ◽  
Asymptotic Validity

It is well known that a graph with m edges can be made triangle-free by removing (slightly less than) m/2 edges. On the other hand, there are many classes of graphs which are hard to make triangle-free, in the sense that it is necessary to remove roughly m/2 edges in order to eliminate all triangles.We prove that dense graphs that are hard to make triangle-free have a large packing of pairwise edge-disjoint triangles. In particular, they have more than m(1/4+cβ) pairwise edge-disjoint triangles where β is the density of the graph and c ≥ is an absolute constant. This improves upon a previous m(1/4−o(1)) bound which follows from the asymptotic validity of Tuza's conjecture for dense graphs. We conjecture that such graphs have an asymptotically optimal triangle packing of size m(1/3−o(1)).We extend our result from triangles to larger cliques and odd cycles.

scholarly journals Optimal packings of edge-disjoint odd cycles

2000 ◽  
Vol 211 (1-3) ◽  
pp. 197-202 ◽  
Author(s):  
Claude Berge ◽  
Bruce Reed
Keyword(s):  
Edge Disjoint ◽  
Odd Cycles

scholarly journals Edge-disjoint odd cycles in graphs with small chromatic number

1999 ◽  
Vol 49 (3) ◽  
pp. 783-786 ◽  
Author(s):  
Claude Berge ◽  
Bruce Reed
Keyword(s):  
Chromatic Number ◽  
Edge Disjoint ◽  
Odd Cycles

scholarly journals Mixed metric dimension of graphs with edge disjoint cycles

2021 ◽  
Vol 300 ◽  
pp. 1-8
Author(s):  
Jelena Sedlar ◽  
Riste Škrekovski
Keyword(s):  
Metric Dimension ◽  
Edge Disjoint ◽  
Disjoint Cycles ◽  
Mixed Metric

scholarly journals Invariants of Stable Maps between Closed Orientable Surfaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 215
Author(s):  
Catarina Mendes de Jesus S. ◽  
Pantaleón D. Romero
Keyword(s):  
Point Of View ◽  
Stable Maps ◽  
Smooth Maps ◽  
Orientable Surfaces

In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine’s Theorem.

scholarly journals On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽  
2021 ◽  
Author(s):  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Polynomial Time ◽  
Disjoint Paths ◽  
Structural Parameter ◽  
Input Graph ◽  
Feedback Vertex Set ◽  
Fpt Algorithms ◽  
Edge Disjoint ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

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