The cyclicity of a graph is the largest integer n for which the graph is contractible to the cycle on n vertices. By analyzing the cycle space of a graph, we establish upper and lower bounds on cyclicity. These bounds facilitate the computation of cyclicity for several classes of graphs, including chordal graphs, complete n-partite graphs, n-cubes, products of trees and cycles, and planar graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 160–170, 1999

http://www.people.vcu.edu/~rhammack/reprints/cyclicity.pdf

Cyclicity of graphs

Contracting Bipartite Graphs to Paths and Cycles

In this paper we prove the following form of the Phragmen-Brouwer Theorem: a locally connected, connected normal T1-space X is unicoherent if and only if for every pair of disjoint nonseparating continua C and D in X, C U D does not separate X. Among the several corollaries is the proposition: X is multicoherent if and only if X is the union of a circular chain of continua { A0, A , A 2, A3} where no three of the A's have a point in common.

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A strong form of the Phragmén-Brouwer theorem

Testing if a given graph $G$ contains the $k$-vertex path $P_k$ as a minor or as an induced minor is trivial for every fixed integer $k\geq 1$. However, the situation changes for the problem of checking if a graph can be modified into $P_k$ by using only edge contractions. In this case the problem is known to be NP-complete even if $k=4$. This led to an intensive investigation for testing contractibility on restricted graph classes. We focus on bipartite graphs. Heggernes, van 't Hof, Leveque and Paul proved that the problem stays NP-complete for bipartite graphs if $k=6$. We strengthen their result from $k=6$ to $k=5$. We also show that the problem of contracting a bipartite graph to the $6$-vertex cycle $C_6$ is NP-complete. The cyclicity of a graph is the length of the longest cycle the graph can be contracted to. As a consequence of our second result, determining the cyclicity of a bipartite graph is NP-hard.

A circular cover of a graph G is a cover fX0; ;Xn 1g of the topological space G by closed connected subsets, indexed over n, with the following properties: Each element in the cover contains a vertex of G, each vertex of G is contained in at most two elements of the cover, and Xa\Xb 6; if and only if b a2 f 1; 0; 1g. The circularity of G is the largest integer n for which there is a circular cover of G with n elements. It is known that the circularity of a planar graph is even. We sharpen this result by proving that the circularity of a plane graph is twice the maximum number of disjoint paths joining two faces of G. This result leads to a polynomial-time algorithm which computes the circularity of any connected planar graph.

/pdf/circularity-of-planar-graphs-3fz7avu3h4.pdf

Circularity of planar graphs

/pdf/circularity-of-graphs-and-continua-topology-3bgie3rz4q.pdf

R. F. Dickamn

Papers

Circularity of graphs and continua: topology