In mathematics, a vertex cycle cover (commonly called simply cycle cover) of a graph G is a set of cycles which are subgraphs of G and contain all vertices of G.
If the cycles of the cover have no vertices in common, the cover is called vertex-disjoint or sometimes simply disjoint cycle cover. This is sometimes known as exact vertex cycle cover. In this case the set of the cycles constitutes a spanning subgraph of G. A disjoint cycle cover of an undirected graph (if it exists) can be found in polynomial time by transforming the problem into a problem of finding a perfect matching in a larger graph.[1][2]
If the cycles of the cover have no edges in common, the cover is called edge-disjoint or simply disjoint cycle cover.
Similar definitions exist for digraphs, in terms of directed cycles. Finding a vertex-disjoint cycle cover of a directed graph can also be performed in polynomial time by a similar reduction to perfect matching.[3] However, adding the condition that each cycle should have length at least 3 makes the problem NP-hard.[4]
^David Eppstein. "Partition a graph into node-disjoint cycles".
^Tutte, W. T. (1954), "A short proof of the factor theorem for finite graphs" (PDF), Canadian Journal of Mathematics, 6: 347–352, doi:10.4153/CJM-1954-033-3, MR 0063008, S2CID 123221074.
In mathematics, a vertexcyclecover (commonly called simply cyclecover) of a graph G is a set of cycles which are subgraphs of G and contain all vertices...
optimization, cyclecover may refer to: VertexcyclecoverVertex disjoint cyclecover Edge cyclecover Other possible meanings include: A cover for a bicycle...
cover have no vertices in common, the cover is called vertex-disjoint or sometimes simply disjoint cyclecover. In this case, the set of the cycles constitutes...
perfect matching) has a vertexcover of size n + k {\displaystyle n+k} . The odd cycle transversal can be transformed into a vertexcover by including both...
Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler...
a cycle double cover. For instance, if a cubic graph contains a triangle, a Δ-Y transform will replace the triangle by a single vertex; any cycle double...
feedback vertex set (FVS) of a graph is a set of vertices whose removal leaves a graph without cycles ("removal" means deleting the vertex and all edges...
Hamiltonian cycle. peripheral 1. A peripheral cycle or non-separating cycle is a cycle with at most one bridge. 2. A peripheral vertex is a vertex whose eccentricity...
Using this method, he showed how to solve the Hamiltonian cycle problem in arbitrary n-vertex graphs by a Monte Carlo algorithm in time O(1.657n); for...
bipartite double cover of G has two vertices ui and wi for each vertex vi of G. Two vertices ui and wj are connected by an edge in the double cover if and only...
edge cover is equal to the size of the maximum independent set, and the size of the minimum edge cover plus the size of the minimum vertexcover is equal...
graph with no Hamiltonian cycle. It is hypohamiltonian, meaning that although it has no Hamiltonian cycle, deleting any vertex makes it Hamiltonian, and...
is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems...
connected undirected graph with no cycles. It is a spanning tree of a graph G if it spans G (that is, it includes every vertex of G) and is a subgraph of G...
{\displaystyle G=(V,E)} , we are to find the minimum number of vertex-disjoint paths to cover each vertex in V {\displaystyle V} . We can construct a bipartite...
belonging to the independent set, forms a minimal vertexcover. That is, the complement is a vertexcover, a set of vertices that includes at least one endpoint...
closely related problem, the feedback vertex set, is a set of vertices containing at least one vertex from every cycle in a directed or undirected graph....
cycle has even length. A simple proof uses the method of switching. Switching a signed graph means reversing the signs of all edges between a vertex subset...
every vertex vi is contained in both ei−1 and ei. The number k is called the length of the cycle. A hypergraph is balanced iff every odd-length cycle C in...
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