On Independent Circuits Contained in a Graph

Erdös, P.; Pósa, L.

doi:10.4153/cjm-1965-035-8

Cited by 232 publications

(177 citation statements)

References 4 publications

(3 reference statements)

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“…(Such a set B is also called a feedback vertex set.) The classical theorem of Erdős and Pósa [5] from 1965 (see for example [2]) states that for each positive integer k there is a positive integer f (k) such that the following holds: for each graph G which does not have k + 1 disjoint cycles (that is, pairwise vertex-disjoint cycles), there is a blocker B of size at most f (k). The least value we may take for f (k) is of order k ln k.…”

Section: Introductionmentioning

confidence: 99%

Random unlabelled graphs containing few disjoint cycles

Kang

McDiarmid

2010

Random Struct Algorithms

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ABSTRACT:We call a set B of vertices in a graph G a blocker if the graph G − B obtained from G by deleting the vertices in B has no cycles. The classical Erdős-Pósa theorem (1965) states that for each positive integer k there is a positive integer f (k) such that in each graph G which contains at most k pairwise vertex-disjoint cycles, there is a blocker B of size at most f (k).We show that, amongst all unlabelled n-vertex graphs with at most k disjoint cycles, all but an exponentially small proportion have a unique blocker of size k. We also determine the asymptotic number and properties of such graphs. For example we study the limiting probability that such graphs are connected, the limiting average number of vertices not in the giant component, and the limiting distribution of the chromatic number. Further, we give more detailed results concerning graphs with no two disjoint cycles.These results complement recent results for labelled graphs by Kurauskas and the second author.

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Section: Introductionmentioning

confidence: 99%

Random unlabelled graphs containing few disjoint cycles

Kang

McDiarmid

2010

Random Struct Algorithms

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“…The number of vertices which break all the circuits, given that every pair of circuits has a common vertex, is an open problem due to Gallai [2, p. 3621. It is likely that this number is 3, as is the case for undirected graphs [3].…”

Section: Now the 3 Circuitsmentioning

confidence: 99%

On independent circuits of a digraph

Kosaraju

1977

Journal of Graph Theory

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If every three circuits of a digraph have a common vertex, then all the circuits have one.Paths and circuits are assumed to be elementary, directed, and of nonzero length. A set of paths is openly disjoint if every pair of paths in the set is vertex disjoint with the possible exception that the starting vertex of one could be the same as the final vertex of the other, if neither path is a circuit. A vertex u whose removal makes a digraph G acyclic is a break vertex of G. A carrier [l, p. 30; 7, p. 171 is a vertex with one in-arc and one out-arc. A digraph is minimally connected if it is a strong digraph and the removal of any arc destroys the strongly connected property. A digraph is arbitrarily traceable from u if each chain obtained by starting from u, and choosing at each vertex u an out-arc of u which has not been previously traversed, is closed and complete, A digraph having a closed complete chain is eulerian.Harary [S] gave a characterization of eulerian digraphs with a break vertex: An eulerian digraph G has a break vertex u if and only if G is arbitrarily traceable from u. Here a characterization of digraphs having a break vertex is given.The following two lemmas are minor variants of well-known results.Lemma 1 [l, p. 311. Every minimally connected digraph has a carrier.

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“…A family F of graphs is said to have the Erdős-Pósa property, if for every integer k there is an integer f (k, F) such that every graph G contains k edge-disjoint subgraphs each isomorphic to a graph in F or a set F of at most f (k, F) edges such that G − F has no subgraph isomorphic to a graph in F. The term Erdős-Pósa property arose because in [5], Erdős and Pósa proved that the family of cycles (without any parity condition) has this property.…”

Section: Importance In Graph Theorymentioning

confidence: 99%

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

Kawarabayashi

Kobayashi

2016

Journal of Combinatorial Theory, Series B

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Finding edge-disjoint odd cycles is one of the most important problems in graph theory, graph algorithm and combinatorial optimization. In fact, it is closely related to the well-known max-cut problem. One of the difficulties of this problem is that the Erdős-Pósa property does not hold for odd cycles in general. Motivated by this fact, we prove that for any positive integer k, there exists an integer f (k) satisfying the following: For any 4-edge-connected graph G = (V, E), either G has edge-disjoint k odd cycles or there exists an edge set F ⊆ E with |F | ≤ f (k) such that G − F is bipartite. We note that the 4-edge-connectivity is best possible in this statement. Similar approach can be applied to an algorithmic question. Suppose that the input graph G is a 4-edgeconnected graph with n vertices. We show that, for any ε > 0, if k = O((log log log n) 1/2−ε ), then the edge-disjoint k odd cycle packing in G can be solved in polynomial time of n. ACM Subject Classification G.2.2 Graph Theory

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On Independent Circuits Contained in a Graph

Cited by 232 publications

References 4 publications

Random unlabelled graphs containing few disjoint cycles

Random unlabelled graphs containing few disjoint cycles

On independent circuits of a digraph

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

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