Exact Algorithms for Finding Longest Cycles in Claw-Free Graphs

Abstract

(2013) 'Exact algorithms for nding longest cycles in claw-free graphs.', Algorithmica., 65 (1). 129 -145. Further information on publisher's website:http://dx.doi.org/10.1007/s00453-011-9576-4Publisher's copyright statement:The original publication is available at www.springerlink.com Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. The Hamiltonian Cycle problem is the problem of deciding whether an n-vertex graph G has a cycle passing through all vertices of G. This problem is a classic NP-complete problem. Finding an exact algorithm that solves it in O

Counting closed trails

“…This improves upon the currently fastest known (randomized) O (1.66 n ) time algorithm for general graphs [1] and the earlier record (deterministic) time O (1.69 n ) for claw-free graphs [5]. To arrive at Corollary 1, we rely on the framework presented by Broersma et al [5, §4] and replace their algorithm for computing optimum closed trails with an algorithm derived from Theorem 1; by an optimum closed trail we refer to a closed trail in a graph H whose vertex set touches (dominates) the maximum number of edges of H .…”

“…[9]; find an Eulerian tour of the optimum closed trail, and form a cycle in cl(G) by traversing the tour and including all touched edges in H into the cycle). Translate in polynomial time the longest cycle in cl(G) into one in G [6,12].…”

“…The present note contributes by yet another statistic, counting closed trails, which enables-via a framework presented by Broersma et al [5]-a faster algorithm for the longest cycle problem on claw-free graphs. A further aim of this note is to popularize the algorithmic trick of "sieving for connectivity" among spanning subgraphs.…”

Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs

“…It is conceivable that these algorithms could be modified to give similar results for constructing longest cycles but the only specific results we know of are an algorithm of Monien [37], subsequently improved by Bodlaender [7] to find a longest cycle in an arbitrary graph G in time O (c(G)! 2 c(G) |G|), and a recent result of Broersma et al [10] which gives an O * (1.8878 |G| ) algorithm for finding a longest cycle when G is claw-free.…”

Theory and application of reciprocal transformation of “path problem” and “time float problem”

“…Some such as the shortest-and longest-path problems, which can be solved using the classic Dijkstra [9] and Bellman [5] algorithms, are easy, and others are very complex. Classic path problems include shortest-path problems such as the traveling salesman problem [3,15,27,37], shortest-path problem with cycles [1,19], fuzzy shortest-path problem [46], stochastic shortest path problem [38], multi-objective shortest-path problem [30], and maximum capacity shortest path problem [6]; longest-path problems such as the longest Hamiltonian cycle problem [7], longest-path problem with cycles [16], probabilistic longest-path problem [28], fuzzy longest-path problem [17], and orderly colored longest path problem [35]; and other path problems such as the multi-constrained path problem [43], optimal path problem in a network [20], k-best path problem [29], minimum k-path connected vertex cover problem [23], and median path problem with bounded length [22]. Problems such as the traveling salesman problem and longest Hamiltonian cycle problem are NP-hard [7,37].…”