The longest simple path problem on graphs arises in a variety of context, e.g., information retrieval, VLSI design, robot patrolling. Given an undirected weighted graph G = (V,E) , the problem consists of finding the longest simple path (i.e., no vertex occurs more than once) onG . We propose in this paper an exact and a tabu search algorithm for solving this problem. We show that our technique...

As observed by Rautenbach and Sereni [SIAM J. Discrete Math. 28 (2014) 335–341] there is a gap in the proof of the theorem of Balister et al. [Combin. Probab. Comput. 13 (2004) 311–317], which states that the intersection of all longest paths in a connected circular arc graph is nonempty. In this paper we close this gap.

In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. As an application, we obtain an interlacing property for the zeros of alternating Eulerian polynomials.

The nth detour chromatic number, χn(G) of a graph G is the minimum number of colours required to colour the vertices of G such that no path with more than n vertices is monocoloured. The number of vertices in a longest path of G is denoted by τ (G) . We conjecture that χn(G) ≤ d τ(G) n e for every graph G and every n ≥ 1 and we prove results that support the conjecture. We also present some suf...

Let lpt(G) be the minimum cardinality of a set of vertices that intersects all longest paths in a graph G. Let ω(G) be the size of a maximum clique in G, and tw(G) be the treewidth of G. We prove that lpt(G) ≤ max{1,ω(G)− 2} when G is a connected chordal graph; that lpt(G) = 1 when G is a connected bipartite permutation graph or a connected full substar graph; and that lpt(G) ≤ tw(G) for any co...

We prove some sufficient conditions for a directed graph to have the property of a conjecture of J. (1982): "Every directed graph contains an independent set which meets every longest directed path". Let G be a directed graph, and denote by V(G) its vertex-set, by A(G) its arc-set, X(G) denotes its chromatic number, and A(G) the length of the longest directed path. Independently, B. Roy and T. ...

We study the max-min paths problem, which represents a game version of the shortest and the longest paths problem in a weighted directed graph. In this problem the vertex set V of the weighted directed graph G = (V, E) is divided into two disjoint subsets VA and VB which are regarded as positional sets of two players. The players are seeking for a directed path from the given starting position ...

We consider the path length in decision diagrams for multiple-valued functions. This is an important measure of a decision diagram, since this models the time needed to evaluate the function. We focus on the average path length (APL), which is the sum of the path lengths over all assignments of values to the variables divided by the number of assignments. First, we show a multiple-valued functi...

A digraph obtained by replacing each edge of a complete mpartite graph with an arc or a pair of mutually opposite arcs with the same end vertices is called a complete m-partite digraph. We describe an O(n) algorithm for finding a longest path in a complete m-partite (m ≥ 2) digraph with n vertices. The algorithm requires time O(n) in case of testing only the existence of a Hamiltonian path and ...

This paper concerns the problem of decomposing a network flow into an integral path flow such that the length of the longest path is minimized.It is shown that this problem isNP-hard in the strong sense.Two approximation algorithms are proposed for the problem:the longest path elimination(LPE)algorithm and the balanced flow propagation(BFP)algorithm.We analyze the properties of both algorithms ...