Circularity of graphs and continua: topology

Cyclicity of graphs

“…The next proposition shows that the computation of the cyclicity of a graph can be reduced to the computation of the cyclicities of its irreducible components. This proposition is a generalization of Theorem 3.6 of [2], applied to cyclicity (rather than circularity).…”

“…Cyclicity was introduced by Blum in [3] as an aid in the study of a related invariant called circularity (see [1,2]). Consequently, it is called co-circularity in [3]; we have renamed it here for clarity, and to emphasize that it is, by itself, an interesting concept.…”

“…If G is a two-connected plane graph and g : G → C n (n ≥ 3) is cyclic, then G has two saturated faces 1. …”

Cyclicity of graphs

“…The next proposition shows that the computation of the cyclicity of a graph can be reduced to the computation of the cyclicities of its irreducible components. This proposition is a generalization of Theorem 3.6 of [2], applied to cyclicity (rather than circularity). Let g : G → C n be a cyclic map.…”

Contracting Bipartite Graphs to Paths and Cycles

“…Of particular relevance is the closely related problem of determining the cyclicity [10] of a graph, that is, the length of a longest cycle to which a given graph can be contracted. Cyclicity was introduced by Blum [3] under the name co-circularity, due to a close relationship with a concept in topology called circularity (see also [1]). Later Hammack [10] coined the current name for the concept and gave both structural results and polynomial-time algorithms for a number of special graph classes.…”