Packing Cycles in Graphs

Abstract

A graph G is called cycle Mengerian (CM) if for all nonnegative integral function w defined on V ðGÞ; the maximum number of cycles (repetition is allowed) in G such that each vertex v is used at most wðvÞ times is equal to the minimum of P fwðxÞ : x 2 X g; where the minimum is taken over all X V ðGÞ such that deleting X from G results in a forest. The purpose of this paper is to characterize all CM graphs in terms of forbidden structures. As a corollary, we prove that if the fractional version of the above minimization problem always have an integral optimal solution, then the fractional version of the maximization problem will always have an integral optimal solution as well. # 2002 Elsevier Science (USA)

An Efficient Algorithm for Finding Maximum Cycle Packings in Reducible Flow Graphs

“…Since x is a maximum r -t flow in G mm (r ), so is x , and hence we have (8). t) by (5), as desired.…”

“…Now we replace x by x and repeat the process. From (2), (8), and (10) we conclude that a good integral maximum r -t flow in G mm (r ) can be obtained after at most n iterations. Since the initial maximum flow x and x in (7) can both be found in O(nm log(n 2 /m)) time [12], the whole algorithm runs in time O(n 2 m log(n 2 /m)).…”

“…vertex) set. See [3]- [5] for a complete characterization of all tournaments and bipartite tournaments and [7] and [8] for the description of all undirected graphs.…”

Erdős-Pósa property of chordless cycles and its applications

“…Our result has an application on the packing and covering for weighted cycles. For a graph G and a non-negative weight functions w : V (G) → N ∪ {0}, let pack(G, w) be the maximum number of cycles (repetition is allowed) such that each vertex v used in at most w(v) times, and let cover(G, w) be the minimum value v∈X w(v) where X hits all cycles in G. Ding and Zang [7] characterized cycle Mengerian graphs G, which satisfy the property that for all non-negative weight function w, pack(G, w) = cover(G, w). Up to our best knowledge, it was not previously known that cover(G, w) can be bounded by a function of pack(G, w).…”

A Min-Max Theorem on Tournaments

“…As conjectured by Edmonds and Giles [9,18] and proved recently by Ding, Feng, and Zang [4], the problem of recognizing Mengerian hypergraphs is NP -hard in general, and hence it cannot be solved in polynomial time unless NP = P . In this paper we study a special class of Mengerian hypergraphs; our work is a continuation of those done in [1,2,3,5,6].…”